Engineering Mathematics Workbook_Final

Complex Variables

105. The value of ‘P’ such that the 109. The value of
function z  cos z dz is

1 1  px   1 = ( z z − 2 )(z − ) 4
2
f ( ) z = log ( x + y 2 ) i+ tan −  
2   y   i  i 
is analytic is (a) (b) −
4 4

(a) 1 (b) -1 (c) i  (d) 2 i 

(c) 2 (d) -2 110. Let ‘C’ be the circle z = 1 in the


sin (z − ) 1 complex plane described in counter
106. The function ( ) z = at z 1 z
+
f
z − 1 clock wise then c  dz
)
−
= 1 is (2 zi z
−
i
(a) removable singular (a) i  (b) 
−
(c) 2 i  (d) 2 i 
(b) essential singular

2

(c) pole of order 2 111. The value of c  sin z + cos z 2 dz
(z − 4 )(z − ) 2
(d) none
where z = 3 is
z − sin z
107. The function ( ) z = at z = (a) 2 i  (b) 2 i 
−
f
z 3
−
i
0 is (c) i  (d) 
(a) removable singular 112. The value of c  sec z dz where ‘C’ is
z = 1 is
(b) essential singular
(a) 0 (b) 2 i 
(c) pole of order 2

(d) none (c) i (d)
2
 1 
108. The function ( ) sinf z =       at c  1
−
 1 z  113. The value of z + 4 dz where ‘c’
2
z = 1 is is

(a) removable singular i 
(a) (b) −
(b) essential singular 2 2
i i
(c) zero of the function (c) − (d)
2 2
(d) none




149

Complex Variables


114. The value of c  e 2z dz where (a) 1 3 − 1 5 + 1 7 − ............
(z + ) 1 4 z z z
‘c’ is z = 3 is (b) 1 + 1 + 1 + ............
z 3 z 5 z 7

8 i  4 i  1 1 1
−
−
2
2
(a) e (b) e (c) + + + ............
3 3 z 2 z 4 z 6
8 i  1 − 1 + 1 −
−
1
(c) e (d) 0 (d) z 2 z 4 z 6 ............
3
118. The Taylor series expansion of
115. The value of  − 3z + 4 dz where f z =  is
( ) sin z about z =
z + 2 4z + 5 4
‘c’ is z = 1 is (a)
1       (z   − / ) 4 2 (z  − / ) 4 3  
1   1+  z −    + + + .... 
(a) 0 (b) 2    4  2! 3!  
10
(b)

4 1       (z   − / ) 4 2 (z  − / ) 4 3  
(c) (d) 1   1+  z −    − − − .... 
5 2    4  2! 3!  

z 3 z 5
116. The Laurent series expansion of (c) z − + − ......
1 3! 5!
f ( ) z = in the valid region
4z − z 2 (d) none
z  4 is 119. The taylor series expansion of

f z = − 1
( ) Tan z about z = 0 is


(a)  z n− 1 (b)  z n− 1 3 5
n i = 4 n+ 1 n= 0 4 n+ 1 z z
(a) z − + − .......
3 5
2n+
n+
1
2


(c)  z (d)  z 3 5
n= 0 4 n+ 2 n= 0 4 2n+ 2 z z
(b) z − − − .......
3 5
117. The Laurent series expansion of
1 z 3 z 5
f ( ) z = , in the region (c) z + + + .......
z (1 z+ 2 ) 3 5
z  1 is z 3 z 5
(d) z + + + .......
3 5







150

Complex Variables

z
f
120. Let ( ) z = . Then z + 2 2z − 1
(a) C  z − 2 dz = 2 i  , where C
( )
(a) lim f z does not exist is a circle z = 3
z→ 0
(b) f is continuous at z = 0 z + 2 2z − 1
(b) C  dz = 14 i  , where
(c) f is not differentiable at z = 0 z − 2
(d) f is not regular at z = 0 C is a circle z = 3


2
121. Let ( ) z = z . Then z + 2 2z − 1
f
(c) C  2 dz = 12 i  , where
(a) Cauchy-Riemann equations are (z − ) 2
satisfied only at z = 0 C is a circle z = 3


(b) Cauchy-Riemann equations are (d) C  z + 2 2z − 1 dz = 4 i  , where C
satisfied for all z (z − ) 2 2

(c) Cauchy-Riemann equations are is a circle z = 3
not satisfied at any values of z

124. Let ( ) z = cos (z − ) 1 , then
f
(d) f is not analytic at z = 0 z − 1
122. Consider the function (a) f(z) has simple pole at z = 1

 e − z − 4 , if z  0 (b) f(z) has isolated essential

f ( ) z =  . Then singularity at z = 1
 0, if z = 0


which of the following (s) is / are (c) residue of at z = 1 is undefined
correct? (d) residue of f at z = 1 is 1

(a) Cauchy-Riemann equations are ze z
f
not satisfied at z = 0 125. Let ( ) z = z − 1 . Then

(b) Cauchy-Riemann equations are C  1
0
satisfied at z = 0 (a) f ( ) z dz = on :C z = 2

(c) f is not analytic at z = 0 1
(b) f(z) is analytic on z =
(d) f is analytic at z = 0 2
1
123. Which of the following (s) is correct? (c) f(z) is analytic within z 
2
1

(d) f(z) is not analytic outside z 
2





151

Complex Variables


( )
126. Let f z = e z be a complex (d) The function f has zeros of order
2 at 2 n , n =  
z 2018 1, 2,...
function. Then
[NET JUNE 2013]
(a) f(z) has a simple pole at z = 0
129. Consider the function
(b) f(z) has a pole at z = 0 of order
( z
2018 sin  / ) 2
f ( ) z = . Then f has
( ) z
(c) The residue of f(z) at z = 0 is sin 
1 poles at
2017! (a) all integers

(b) all even integers
(d) The residue of f(z) at z = 0 is
1 (c) all odd integers

2018! (d) all integers of the form 4k + 1,

Z
127. The value of the contour integral k 
2
→
  z − 1 dz is 130. Let f :C C be a holomorphic
z function and let u be the real part of f
(a) 0, when the curve is a and  be the imaginary of f. Then,
semicircle z = 2e i (0     ) for , x y  , f ( ' x  ) y 2 is equal
+
R
(b) − i , when the curve  is a to
)
 
semicircle z = 2e i (   (a) u + u (b) u + u
2
2
2
2
x
x
y
y
(c) − i , when the curve  is a (c) u + u (d) u − u
2
2
2
2
semicircle z = 2e i (   2 ) y y x y

131. Let f(z) be the meromorphic function
−
(d) 2i , when the curve  is a z
f
semicircle z = 2e i (0    2  ) given by ( ) z = . Then
−
(1 e z )sin z
128. Consider the function
)

−
f ( ) z = z 2 (1 cos z , z C . Which (a) z = 0 is a pole of order 2.

Z
of the following are correct? (b) For every k  , z = 2 ik is a
simple pole.
(a) The function f has zeros of order 2
=
k
at 0 (c) For every k  / z   0 , z  is a
(b) The function f has zeros of order simple pole.


1 at 2 n , n = 1, 2,..... (d) z  = 2 i is a pole.
 +
(c) The function f has zeros of order 4
at 0.

152

Complex Variables
−
− −
1 (a) 1, 1, 1

132. Let ( ) z = for all z C such
f
e − 1 (b) 1, , 2
z
that e  1. Then 
z
− + +
2
+
(a) f is meromorphic (c) 1 1 2 , 1 2
(b) the only singularities of f are (d) 1,1 2 ,1 2− −  − 2
poles.
137. Which of the following is possible
(c) f has infinitely many poles on the value for the imaginary part of
imaginary axis ln ( )
i

(d) Each pole of f is simple.

z − 1 (a)  (b)
133. Let ( ) z =   then, 2
f
exp   2 i      − 1 
 z  (c) (d) 
4 8
(a) f has an isolated singularity at
z = 0 [GATE]

(b) f has a removable singularity at 138. e is a periodic with a period of
z
z=1

(c) f has infinitely many poles (a) 2 (b) 2 i 

(d) each pole of f is of order 1. (c)  (d) i

134. If z − 1 = 2 , then zz − − = ____ [GATE-1997-CE]
z
z
139. Which one of the following is not
135. Given to complex numbers true for complex number z and z ?

5 3 i and z =
z = 5 + ( ) 2 + 2i 1 2
1
2
3 z z z
(a) 1 = 1 2
z z 2
the argument of 1 in degree is 2 z 2
z 2 (b) z + z  z + z
2
1
2
1
(a) 0 (b) 30
(c) z − z  z − z
2
2
1
1
(c) 60 (d) 90
(d) z + z 2 + z − z 2 = 2 z 2 + 2 z 2
136. If 1, ,  2 are cube roots of units, 1 2 1 2 1 2
3
then the roots of (x − ) 1 + 8 0 are [GATE-2005-CE]
=




153

Complex Variables

140. Consider the circle z − − 2 (a) 1 (b) e x − y 2
2
5 5i = in
the complex number plane (x, y) with (c) e (d) e − y
y
z = x iy . The minimum distance
+
from the origin to the circle is [GATE-2009 (IN)]
3
j
145. One of the roots of equation x = ,
2
(a) 5 2 − (b) 54
where j is the positive square roots of
(c) 34 (d) 5 2 -1 is
[GATE-2005 (IN)] 3 j

3
141. Let z = , where z is a complex (a) j (b) 2 + 2
z
number not equal to zero. Then z is a 3 j 3 j
solution of (c) − (d) − −
2 2 2 2
2
3
(a) z = 1 (b) z = 1 [GATE-2009 (IN)]
4
9
(c) z = 1 (d) z = 1
146. If x = − 1 , then the value of x is
x
[GATE-2005 (IN)]
(a) e −  /2 (b) e  /2
3 1
142. If a complex number z = + i (c) x (d) 1
2 2
4
then z is [GATE-2012-EC, EE, IN]
147. Square roots of -i, where i = − 1,
1 3
(a) 2 2 + 2i (b) + i are
2 2
3 1 3 1 (a) i, -1
(c) i − (d) i −      


2 2 8 8 (b) cos −  i + sin −  ,
 
 
 
 
[GATE-2007 (PI)]  4   4 
 3   3 
143. The equation sin (z) = 10 has cos   i + sin  
 
 
 
 
 4   4 
(a) no real (or) complex solution     3  
(c) cos   i + sin   ,
 
 
 
 
(b) exactly two distinct complex  4   4 
solution  3    
cos   i + sin  
 
 
 
 
(c) a unique solution  4   4 
(d) an infinite number of complex  3   3 




solutions (d) cos   4   i + sin −   4   ,




[GATE-2008 (ME)]  3    3  
cos −   i + sin  
 
 
 
 
144. If Z = x + jy where x, y are real  4   4 
when the value of e jz is [GATE-2013-EE]

154

Complex Variables

148. All the values of the multi-valued GRAPH OF COMPLEX FUNCTION
complex function 1 , where i = − 1, 2
i
f
are 152. The function ( ) z = z maps first
(a) purely imaginary quadrant onto _____
(b) real and non-negative
(a) itself
(c) on the unit circle (b) upper half plane
(d) equal in real and imaginary parts

[GATE-2014 – EE-SET 2] (c) third quadrant
(d) right half plane
149. Given two complex number

5 3 i and z =
z = 5 + ( ) 2 + 2i, 153. The bilinear transformation
2
1
3 z − 1
z w = z + 1
the argument of 1 in degree is
z
2
(a) maps the inside of the unit circle
(a) 0 (b) 30 in the z-plane to the left half of the w-
(c) 60 (d) 90 plane
[GATE-2015-ME-SET 1]
(b) maps the outside the unit circle in
150. Which one of the following options
correctly describes the location of the the z-plane to the left half of the w-
4
2
roots of the equation s + s + = plane
1 0
on the complex plane? (c) maps the inside of the unit circle
(a) Four left half plane (LHP) roots in the z-plane to right half of the w-

(b) One right half plane (RHP) root, plane
one LHP root and two roots on the
imaginary axis (d) maps the outside the unit circle in
the z-plane to the right half of the w-
(c) Two RHP roots and two LHP plane
roots

[GATE-2002 (IN)]
(d) All four roots are on the
imaginary axis 154. For the function of a complex
variable W = ln Z (where, W = u + jv
[GATE-2017 EC SESSION-1]
and

151. Let z = x + jy where j = − 1 . Z = x + jy), then u = constant lines

Then cos z = get mapped in z-plane as

(a) set of radial straight lines
(a) cos z (b) cos z
(b) set of concentric circles
(c) sin z (d) sin z

[GATE-2017 (IN)]



155

Complex Variables

(c) set of confocal hyperbolas
(d) set of confocal ellipses

[GATE-2006-EC] (d)


155. A complex variable z = x + j(0.1) has
its real part x varying in the range
− to . Which one of the following
is the locus (shown in thick lines) of [GATE-2008 (IN)]
1
in the complex plane? 156. A point z has been plotted in the
z
complex plane, as shown in figure
below






(a)









1
lies in the curve
z


(b)




(a)








(c)



(b)











156

Complex Variables

1 − 1  px  
(
2
f ( ) z = log x + y 2 ) + i tan    
2  y 
is analytic is ______. [JNU]
(c)
159. If ( ) ( x + ay 2 ) + ibxy is
f z =
2
complex analytic function of

z = x iy , where i = − 1, then
+

(a) a = − 1, b = − 1

(b) a = − 1, b =
2
(d)
(c) a = 1, b = 2


(d) a = 2, b = 2 [GATE 2017]

)
)


160. If ( , x y and ( , x y are functions
[GATE-2011-EE]
with continuous second derivatives,
)
157. Let S be the set of points in the then ( ,x y ) i + ( ,x y can be
complex plane corresponding to the expressed as an analytic function of

unit circle. That is S =    : z z = 1 . x iy i = − 1 when
(
)


+
Consider the function f(z) = zz’where
z’ denotes the complex conjugate of        
z. The f(z) maps S to which one of (a) = , =
the following in the complex plane  x  x  y  y
(a) unit circle (b)   =     =  
,
 y  x  x  y
(b) horizontal axis line segment from
origin to (1, 0)  2   2   2   2 
(c) 2 + 2 = 2 + 2 = 1
(c) the point (1, 0)  x  y  x  y


(d) the entire horizontal axis   +   =   +   =
(d)  x  y  x  y 0
[GATE-EE-SET 1]
[GATE-2007-CE]
CAUCHY-REIMANN EQUATIONS
161. Consider the complex valued
158. The value of ‘P’ such that the function ( ) 2f z = z + 3 3
function b z where z
is a complex variable. The value of b



157

Complex Variables

for which the function f(z) is analytic (c) 2x + x +
−
is ______. x + y 2 C
2
[GATE-2016-EC-SET 2] y
(d) 2xy − + C
2
162. What is value of the m for which x + y 2
2x x + 2 my is harmonic?
2
−
[ESE 2017 (COMMON PAPER)]
(a) 1 (b) -1
165. The value of  zdz from z = 0 to z =

(c) 2 (d) -2 c
4 + 2i along the curve ‘c’ given by
=
[ESE 2017 (EE)] z t + 2 it
CONSTRUCTION OF AN ANALYTIC 8i 8
FUNCTION (a) 10 − (b) 10i +
3 3

163. The real part of an analytic function 8
f(z) where z = x + jy is given by (c) 10 − (d) 0
3i
e − y cos ( ) x . The imaginary part of
166. If z is a complex variable, the value
f(z) is 3i
of  dz
−
(a) e y cos ( ) x (b) e y sin ( ) x 5 z

(c) e − y sin ( ) x (d) e − − y sin ( ) x (a) -0.511 – 1.57i

(b) – 0.511 + 1.57i
[GATE-2014-EC-SET 2]
(c) 0.511 – 1.57i
164. If W  = i + represents the
complex potential for an electric (d) 0.511 + 1.57i
field.
[GATE-2014-ME-SET 1]
x
2
2
Given  = x − y + , then
x + y 2 167. Consider the line integral
2
)
2
the function  is I = (  x + 2 iy dz , where z = x +
c
y iy. The line c is shown in the figure
(a) 2x− + + C below.
2
x + y 2
x
(b) 2xy + + C
2
x + y 2







158

Complex Variables

COMPLEX INTEGRATION USING
CAUCHY RESDUE THEOREM

i
169. Let  = e , then residue of
10
1
f ( ) z = at z = is
+
1 z 10
The value of I is
 
1 2 (a) − (b)
(a) i (b) i 10 10
2 3
i −  i  
3 4 (c) (d)
(c) i (d) i 5 5
4 5
sin z
170. The residue of ( ) z = at z = 0
f
[GATE-2017 EE SESSION-I] 8
z
COMPLEX INTEGRATION USING is
CAUCHY INTEGRAL 1
THEOREM (a) 0 (b) −
7!
168. Consider likely applicability of
Cauchy’s integral theorem to evaluate (c) 1 (d) none

the following integral counter 7!
clockwise around the unit circle C, [GATE]
I =  sec zdz z being a complex
,
−
c 171. If ( ) z = c + c z , then
1
f
variable. The of I will be 0 1
 1+ f ( ) z dz is given by
(a) I = 0 singularities set = 
unit z
circle
(b) I = 0 singularities set
+
  2n + 1   (a) 2 C (b) 2 (1 C )

=   , n = 0,1,2.... 1 0

  2  

+

(c) 2 jC (d) 2 j (1 C 0 )
1

(c) I = , singularities set [GATE-2009-EC]
2
 n
=  ; n = 0,1,2....  172. If C is a circle of radius r with centre
z , in the complex z-plane and if n is
0
(d) None of the above

[GATE-2005-CE]




159

Complex Variables

a non-zero integer, then (a) the residue of z at z = 1 is
2
C  dz equals z − 1
(z − z 0 ) nt 1/2

2

(a) 2 n j (b) 0 (b) C  z dz = 0

nj 1 1

(c) (d) 2 n (c) C  dz = 1
2 2 z

[GATE-2015-EC-SET 3] (d) z (complex conjugate of z) is an
analytical function.
173. The value of  dz where C is
+
C ( 1 z 2 ) [GATE-2015-EC-SET 1]

i 176. In the following integral, the contour
the contour z − = 1 is C encloses the points 2 j and

2

− 2 j .
(a) 2 i  (b)  1 sin z
−
−
1
1
(c) tan z (d) i  tan z − 2 C  (z − 2 ) j 3 dz
[GATE-2007-EC]
The value of the integral is ………….
z
174. Given ( ) z = with z  , [GATE-2016]
a
X
−
(z a ) 2 dz
=
X
the residue of ( ) z z n− 1 at z a for 177. Evaluate  z sin z , where c is
n = 0 will be x + 2 y = 2 1

n
(a) a n− 1 (b) a (a) 1 (b) 2

n
(c) n a (d) n a n− 1 (c) 0 (d) -1
[ESE 2017 (EE)]
[GATE-2008 (EE)]
178. The countour C given below is on the
175. Let z = x + iy be a complex variable. complex plane z = x + jy, where
Consider that contour integration is j = − 1 .

performed along the unit circle in
anticlockwise direction. Which one of The value of the integral
the following statements is NOT 1 dz
TRUE?  C  z − 1 is ___________
2

[GATE 2018 (EC)]




160

Complex Variables

179. What is the residue of the function 182. The contour integral  e dz with C
1
z
−
1 e 2z at its pole? C
z 4 as the counter clock wise unit circle
in the z-plane is equal to
4 4
(a) (b) −
3 3 (a) 0 (b) 2

2 2 1
(c) − (d) (c) 2 − (d) 
3 3
[GATE-2011 (IN)]
[ESE 2018 (COMMON PAPER)]
183. In the Laurent expansion of
LAURENT EXPANSION 1
f ( ) z = valid in the
180. The Taylor series expansion of f(z) = (z − 1 )(z − ) 2

sin z about z = is region 1 z  2 , the coefficient of
4
1
   2   3  is
   z −    z −     z 2
1       4     4   
(a) 1+     z −     +   +   + .....  
2   4  2! 3!  1
  (a) 0 (b)
    2

   2   3 
   z −    z −     (c) 1 (d) -1
1        4       4    
(b) 1+     z −     − − + .....  
2   4  2! 3!  [ESE 2018 (COMMON PAPER)]
 
     
1
184. The coefficient of in the laurent
z 3 z 5 z
(c) z − + − .......
3! 5!  z 
series expansion of log       valid
(d) none [GATE]  z − 1 
in z  1 is _____ [GATE]
sin z
181. For the function of a complex
z 3 185. In Laurent series expansion of

variable z, the point z = 0 is 1 1
f ( ) z = − valid in the
(a) a pole of order 3 z − 1 z − 2
1
2
(b) a pole of order 2 region z  , the coefficient of
z 2
(c) a pole of order 1 is _________. [GATE]

(d) not a singularity


[GATE-2007 (IN)]



161

6















Probability














&

















Statistics

Probability & Statistics

1. Suppose A and B are two 4. Two players, A and B, alternately
independent events with probabilities keep rolling a fair dice. The person to
P A  P B  get a six first wins the game. Given
( ) 0 and ( ) 0 . Let A and
that player A starts the game, the
B be their complements. Which one probability that A wins the game is
of the following statements is
FALSE? 5 1
(a) 11 (b) 2

P
(a) ( A B = ) P A P B
( ) ( )
7 6
)
P
(b) ( | A B = P ( ) A (c) 13 (d) 11
)

P
( )
(c) ( A B = P ( ) A + P B [GATE-2015-EE-SET-1]
( ) ( )
(
)

(d) P A B = P A  P B 5. Ram and Ramesh appeared in an
interview for two vacancies in the
same department. The probabilities of
[GATE-2015-EC-SET-1]
Ram’s selection is 1/6 and that of
2. Three vendors were asked to supply a Ramesh is 1/8. What is probability
very high precision component. The that only one of them will be

respective probabilities of their selected.
meeting the strict design (a) 47/48 (b) 1/4
specifications are 0.8, 0.7 and 0.5.
Each vendor supplies one component. (c) 13/48 (d) 35/48
The probability that out of total three
components supplied by the vendors [GATE-2015-EC-SET-2; ME - SET-1]
at least one will meet the design 6. Two coins R and S are tossed. The 4

specifications is __________. joint events H H , T T , T H
S
R S
R
S
R
[GATE-2015] have probabilities 0.28, 0.18, 0.30,
0.24 respectively, where H represents
3. If P(X) = 1/4, P(Y) = 1/3 and head and T represents tail. Which one
P ( X  Y ) 1/12= , the value of of the following is TRUE?
P(Y/X) is
(a) The coin tosses are independent

1 4
(a) (b) (b) R is fair, S is not
4 25
(c) S is fair, R is not
1 29
(c) (d) (d) The coin tosses are dependent
3 50

[GATE-2015-EE-SET-2]
[GATE-2015-ME-SET-3; 1 MARK]



163

Probability & Statistics
)
−
−
7. A coin is tossed thrice. Let X be the (a) pq + (1 p )(1 q
event that head occurs in each of the
first two tosses. Let Y be the event (b) pq
that a tail occurs on the third toss. Let (c) (1p − ) q
Z be the event that two tails occur in
−
three tosses. Based on the above (d) 1 pq
information which one of the
following statements is TRUE? [GATE-2015-CS-SET-2]

(a) X and Y are not independent 10. The chance of a student passing an
exam is 20%. The chance of student
(b) Y and Z are dependent passing the exam and getting above
90% marks in it is 5%. Given that a
(c) Y and Z are independent
student passes the examination, the

(d) X and Z are independent probability that the student gets above
90% marks is
[GATE-2015]
1 1
8. The input X to the Binary Symmetric (a) (b)
Channel (BSC) shown in the figure 18 4

‘1’ with probability 0.8. The 2 5
crossover probability is 1/7. If the (c) 9 (d) 18
received bit Y = 0, the condition
probability that ‘1’ was transmitted is [GATE-2015-ME-SET-2]
________. 11. A product is an assemble of 5

different components. The product
can be sequentially assembled in two
possible ways. If the 5 components
are placed in a box and these are
drawn at random from the box, then

the probability of getting a correct
sequence is

2 2
(a) (b)
[GATE-2015-CS-SET-1] 5! 5
2 2
0,1
0,1
9. Let X  ( ) and Y  ( ) be two (c) (d)
independent binary random variables. (5 2− )! (5 3− )!
If ( X = ) 0 = p and (Y = ) 0 = q, [GATE-2015 (PI) ]
P
P
then ( X + Y  ) 1 is equal to
P




164

Probability & Statistics

12. Given Set A = {2, 3, 4, 5} and Set B 14. Four cards are randomly selected
= {11, 12, 13, 14, 15}, two numbers from a pack of 52 cards. If the first
are randomly selected one from each two cards are kings, what is the
sell. What is the probability that the probability that third card is a king?
sum of the two numbers equals 16?
4
(a)
(a) 0.20 (b) 0.25 52

(c) 0.30 (d) 0.33 2
(b)
[GATE-2015 (EE-SET-1 & CS-SET-1)] 50

13. The probabilities that a student passes  1    1 




in Mathematics, Physics and (c)   52     52  




Chemistry are m, p and c
respectively. Of these subjects, the (d)  1    1    1 






student has 75% chance of passing in   52     51     50  






at least one, a 50% chance of passing
in at least two and a 40% chance of [GATE-2015-CE-SET-2]
passing in exactly two. Following 15. The probability of getting a “head” in
relations are drawn in m, p, c
a single toss of a biased coin is 0.3.
27 The coin is tossed repeatedly till a
+
+
I. p m c = “head” is obtained. If the tosses are
20
independent, then the probability of
13 getting “head” for the first time in the
+
+
II. p m c =
20 fifth toss is _________.
1 [GATE-2016-EC-SET-3]
p 
m 
III. ( ) ( ) ( ) c =
10 16. Candidates were asked to come to an
interview with 3 pens each. Black,
(a) Only relation I is true
Blue, green and red were the
(b) Only relation III is true permitted pen colours that the
candidate could bring. The
(c) Relations II and III are true
probability that a candidate comes
(d) Relations I and III are true with all 3 pens having the same
colour is _________.
[GATE-2015 (EE-SET-1 & CS-SET-1)]
[GATE-2016-EE-SET-1]








165

Probability & Statistics

17. X and Y are two random independent probability of getting a red ball in the
events. It is known that P(X) = 0.40 next draw is
and ( X  Y C ) 0.7. Which one of
=
P
the following is the value of (a) 65 (b) 67
P ( X  Y ) ? 156 156
79 89
(a) 0.7 (b) 0.5 (c) (d)
156 156
(c) 0.4 (d) 0.3
[GATE-2016 (IN)]
[GATE-2016-CE-SET-2]
21. A fair coin is tossed N times. The
18. Three cards were drawn from a pack probability that head does not turn up
of 52 cards. The probability that they in any of the tosses is
are a king, a queen, and a jack is
 1  N 1 −  1  N 1 −
16 64 (a)   (b) −  
(a) (b)   2     2  
5525 2197    


3 8   1   N   1   N
(c) (d) (c)     (d) 1−    
13 16575  2   2 


[GATE-2016] [GATE-2016 (PI)]

19. A person moving through a 22. Consider the following experiment:
tuberculosis prone zone has a 50%
probability of becoming infected. Step-1: Flip a fair coin twice.
However, only 30% of infected Step-2: If the outcomes are (TAILS,
people developed the disease. What HEADS) the output is Y and stop.
percentage of people moving through
a tuberculosis prone zone remains Step-3: If the outcomes are either
infected but does not show symptoms (heads, heads) or (HEADS, TAILS),
of disease? then output is N and stop.

(a) 15 (b) 33 Step-4: If the outcomes are (TAILS,
TAILS), then go to Step-1.
(c) 35 (d) 37
The probability that the output of the
[GATE-2016]
experiment is Y is (upto two decimal
20. An urn contains 5 red and 7 green places) ________
balls. A ball is drawn at random and [GATE-2016-CS-SET-1]
its colour is noted. The ball is placed
back into the urn along with another 23. Suppose that a shop has an equal

ball of the same colour. The number of LED bulbs of two
different types. The probability of an
LED bulb lasting more than 100



166

Probability & Statistics

hours given that it is of type 1 is 0.7, 27. Two coins are tossed simultaneously.
and given that it is of type 2 is 0.4. The probability (upto the decimal
The probability that an LED bulb points accuracy) of getting at least
chosen uniformly at random lasts one head is __________.
more than 100 hours is

____________. [GATE-2017-ME-SESSION-2]
28. The number of integers between 1
[GATE-2016-CS-SET-2] and 500 (both inclusive) that are

divisible by 3 or 5 or 7 is
24. Three fair cubical dice are thrown
simultaneously. The probability that ___________. [GATE-2017]
all three dice have the same number 29. The probability that a k-digit code
on the faces showing up is (up to does NOT contain the digits 0, 5 or 9
third decimal place) _____________. is

[GATE-2017-EC-SESSION-1] k k
(a) 0.3 (b) 0.6
25. An urn contains 5 red balls and 5 k k
black balls. In the first draw, one ball (c) 0.7 (d) 0.9
is picked at random and discarded [GATE-2017 - EE]
without noticing its colour. The
probability to get a red ball in the 30. The probability that a communication
second draw is system will have high fidelity is 0.81.


1 4 The probability that the system will
(a) (b) have both high fidelity and high
2 9
selectivity is 0.18. The probability
5 6 that a given system with high fidelity
(c) (d)
9 9 will have high selectivity is

[GATE-2017-EE-SESSION-2] (a) 0.181 (b) 0.191

26. A two-faced fair coin has its faces (c) 0.222 (d) 0.826
designated as head (H) and tail (T).
This coin is tossed three times in [GATE-2017 (IN)]
succession to record the following
outcomes: H.H.H. If the coin is 31. Two dice are thrown simultaneously.
tossed one more time, the probability The probability that the product of
(up to one decimal place) of the numbers appearing on the top
obtaining H again given the previous faces of the dice is a perfect square is
realisations of H, H and H. would be
________ (a) 1/9 (b) 2/9

[GATE-2017-CE-SESSION-2] (c) 1/3 (d) 4/9


[GATE-2017-IN]




167

Probability & Statistics

32. A box has 6 red balls and 4 while 35. A bag contains 7 red and 4 white
balls. A ball is picked at random and balls. Two balls are drawn at random.
replaced in the box, after which a What is the probability that both the
second balls is picked. balls are red?
28 21
The probability of both the balls (a) 55 (b) 55
being red, rounded to 2 decimal
places, is ________. 7 4
(c) (d)
[GATE-2017-CH] 55 55
[ESE – 2017 - EE]
33. P and Q are considering to apply for a
job. The probability that P applies for 36. Probability (up to one decimal place)
of consecutively picking 3 red balls
1
the job is . The probability that P without replacement from a box
4 containing 5 red balls and 1 white

applies for the job given that Q ball is _________.
1
applies for the job is , and the [GATE-2018-CE-AFTERNOON
2 SESSION]

probability that Q applies for the job 37. An unbiased coin is tossed six times
1 in a row and four different such trials
given that P applies for the job is .
3 are conducted.
Then the probability that P does not One train implies six tosses of the
apply for the job given that Q does coin. If H stands for head and T

not apply for the job is stands for tail, the following are the
observations from the four trials.
4 5
(a) (b) (1) HTHTHT (2) TTRHHHT
5 6
(3) HTTHHT (4) HHHT___
7 11
(c) (d)
8 12 Which statement describing the last
two coin tosses of the fourth trial has
[GATE-2017-PAPER-2-CS] the highest probability of being
correct?
34. There are 3 red socks, 4 green socks
and 3 blue socks, you choose 2 socks. (a) Two T will occur
The probability that they are of the (b) One H and one T will occur
same colour is ________.
(c) Two H will occur
(a) 1/5 (b) 7/30 (d) One H will be followed by one T.


(c) 1/4 (d) 4/15 [GATE-2019-ME-AFTERNOON

SESSION]
[GATE-2017-PAPER-2-CS]


168

Probability & Statistics

38. A c ab was involved in a hit and run 40. The probability density function of a
accident at night. You are given the random variables x is
following data about the cabs in the x − 2

2 0
city and the accident. f ( ) x = 4 (4 x ) ; for 0 x  =
; otherwise
(i) 85% of cabs in the city are
green and the remaining cabs are The mean  of the random variable
blue. k
is __________
(ii) A witness identified the cab [GATE-2015-CE-SET-2, EE-SET-1, CS-
involved in the accident as blue.
SET-1]

(iii) It is known that a witness can 41. Consider the following probability
correctly identify the cab colour only mass function (p.m.f) of a random
80% of the time.
variable X.

Which of the following options is  q if X = 0
closest to the probability that the 
 
)
−
accident was caused by a blue cab? p ( , x q =  1 q if X = 1

 0 otherwise

(a) 12% (b) 15% 
If q = 0.4, the variance of X is
(c) 41% (d) 80%
_________
[GATE-2018-EC]
[GATE-2015-CE-SET-1]
39. A class of twelve children has two
more boys, than girls. A group of 42. The probability of obtaining at least
three children are randomly picked two ‘Six’ in throwing a fair dice 4
from this class to accompany the times is
teacher on a field trip. What is the (a) 425/432 (b) 19/144
probability that the group
accompanying the teacher contains (c) 13/144 (d) 125/432

more girls than boy?
[GATE-2015-ME-SET-1]
325
(a) 0 (b) 43. Let the random variable X represent
864 the number of times a fair coin needs

to be tossed till two consecutive
525 5
(c) (d) heads appear for the first time. The
864 12
expectation of X is _________.

[GATE-2018-EE] [GATE-2015]







169

Probability & Statistics

44. A fair die with faces {1, 2, 3, 4, 5, 6} value of the function ( ) x = e 3 /4 is
x
g
x
is thrown repeatedly till ‘3’ is _________. [GATE-2015-IN]
observed for the first time. Let X
denote the number of times the die is 49. Probability density function of a
thrown. The expected value of X is random variable x is given below:

_________.
 0.25 1 x  5
if 

[GATE-2015-EC-SET-3] f ( ) x =  0 otherwise


45. The variance of the random variable
X with probability density function P (x  ) 4 is

1 −
f ( ) x = x e x is ________. 3 1
2 (a) (b)
4 2

[GATE-2015-CS-SET-3]
1 1
(c) (d)
46. A random variable X has probability 4 8
density function f(x) as given below:
[GATE-2016-CE-SET-1]
 a bx for  1
0 x 
+

f ( ) x =  50. The spot speed (expressed in km/hr)
 0 otherwise observed at a road section are 66, 62,

45, 79, 32, 51, 56, 60, 53 and 40. The
If the expected value E[X] = 2/3, then
Pr[X < 0.5] is __________. median speed expressed in km/hr is
__________.
[GATE-2015-CS-SET-1]
[GATE-2016-CE-SET-2]
47. The probability that a thermistor

randomly picked up from a produced 51. A normal random variable X has the
unit is defective is 0.1. The following probability density function
probability that out of 10 thermistors   (x− ) 1   2  
randomly picked up, 3 are defective f ( ) x = 1 e −   8      ,− x
is x 8


(a) 0.001 (b) 0.057 −
Then 1  f x ( ) x dx ?
(c) 0.107 (d) 0.3
1
(a) 0 (b)
[GATE-2015-IN] 2

48. The probability density function of a 1
−
random variable X is ( ) x = e for (c) 1− (d) 1
x
P
e
x
x  0 and 0 otherwise. The expected
[GATE-2016 (PI)]

170

Probability & Statistics

52. The probability density function on 57. Let the probability density function of
2
the interval [a, 1] is given by 1/ x a random variable, X be given as:
and outside this interval the value of
the function is zero. The value of a is f ( ) x = 3 e u ( ) x + ae u −
−
3x
4x
( ) x
_____. x 2
[GATE-2016-CS-SET-1] where u(x) is the unit step function.
53. The second moment of a Poisson- Then the value of ‘a’ and Prob
distributed random variable is 2. The
mean of the random variable is X   0 , respectively. Are
___________
1 1
[GATE-2016] (a) 2, (b) 4,
2 2
54. Consider a Poisson distribution for
1
1
the tossing of a biased coin. The (c) 2, (d) 4,
mean for this distribution is  . The 2 4
standard deviation for this
distribution is given by [GATE-2016-EC-SET-3]

58. If f(x) and g(x) are two probability
2
(a)  (b) 
density functions,
(c)  (d) 1/ 

[GATE-2016-ME-SET-1]  x
x

55. Two random variables X and Y are  a + 1: − a   0

distributed according to f ( ) x =  x

x a

 − + 1: 0  
a

 (x + y ), 0   1,0   1 
y
x

)
x
f xy ( ,x y =  0, otherwise  − a : −   0
x
a



  x
 
x a
g ( ) x =  : 0  

P
The probability ( X + Y  ) 1 is __.  a
 0: otherwise

[GATE-2015-EC-SET-2] 


56. The area (in percentage) under
standard normal distribution curve of Which of the following statements is
true?
random variable Z within limits from
-3 to +3 is __________.
[GATE-2016-ME-SET-1]

171

Probability & Statistics

(a) Mean of f(x) and g(x) are same; successfully make exactly 6 free
Variance of f(x) and g(x) are same. throws in 10 attempts?

(b) Mean of f(x) and g(x) are same; (a) 0.2508 (b) 0.2816
Variance of f(x) and g(x) are

different. (c) 0.2934 (d) 0.6000
[GATE-2016]
(c) Mean of f(x) and g(x) are
different; Variance of f(x) and g(x) 62. Passengers try repeatedly to get a seat
are same. reservation in any train running

between two stations until they are
(d) Mean of f(x) and g(x) are
different; Variance of f(x) and g(x) successful. If there is 40% chance of
are different. getting reservation in any attempt by
a passenger then the average number

[GATE-2016-CE-SET-2] of attempts that passengers need to
make to get a seat reserved is _____.
59. The probability that a screw
manufactured by a company is [GATE-2017-EC-SESSION-2]
defective is 0.1. The company sells
screws in packets containing 5 screws 63. A person decides to toss a fair coin

and gives a guarantee of replacement repeatedly until he gets a head. He
if one or more screws in the packet will make at most 3 tosses. Let the
are found to be defective. The random variable Y denote the number
probability that a packet would have of heads. The value of var(Y), where
to be replaced is ________. var(.) denotes the variance,

7 49
[GATE-2016-ME-SET-2] (a) (b)
8 64
60. Let X , X and X be independent
1 2 3
and identically distributed random (c) 7 (d) 105
variables with the uniform 64 64
distribution on [0, 1]. The probability [GATE-2017-EE-SESSION-2]
P ( X + 1 X  2 X 3 ) is largest is

_______. 64. Consider the random process
+
[GATE-2016-CE-SET-1] X ( ) t = U Vt


61. Shaquille O Neal is a 60% career free where U is a zero-mean Gaussian
throw shoots, meaning that he random variable and V is a random
successfully makes 60 free throws out variable uniformly distributed
of 100 attempts on average. What is between 0 and 2. Assume that U and
the probability that he will V are statistically independent. The



172

Probability & Statistics

+
mean value of the random process at t 68. For the function ( ) x = a bx ,
f
= 2 is __________. 0 x  1, to be a valid probability

density function, which one of the
[GATE-2017-EC-SESSION-2]
following statements is correct?
65. Assume that in a traffic junction, the (a) a = 1, b = 4 (b) a = 0.5, b = 1
cycle of the traffic signal lights is 2
minutes of green (vehicle does not (c) a = 0, b = 1 (d) a = 1, b = -1
stop) and 3 minutes of red (vehicle [GATE-2017-CE-SESSION-1]
stops). Consider that the arrival time
of vehicles at the junction is 69. For a construction project the mean
and standard deviation of the
uniformly distributed over 5 minute completion time are 200 days and 6.1
cycle. The expected waiting time (in days respectively. Assume normal
minutes) for the vehicle at the distribution and use the value of
junction is _________ standard normal deviate z = 1.64 for
the 95% confidence level. The
[GATE-2017-EE-SESSION-2] maximum time required (in days) for

the completion of the project would
66. The number of parameters in the be _____________
univariate exponential and Gaussian
distributions, respectively are [GATE-2017-CE-SESSION-2]

70. A six-face fair dice is rolled a large
(a) 2 and 2 (b) 1 and 2
number of times. The mean value of
(c) 2 and 1 (d) 1 and 1 the outcomes is _______.

[GATE-2017-ME-SESSION-1]
[GATE-2017-CE-SESSION-1]
71. A sample of 15 data is as follows 17,
67. Vehicles arriving at an intersection 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20,
from one of the approach roads 17, 3. The mode of the data is _
follow the Poisson distribution. The
mean rate of arrival is 900 vehicles (a) 4 (b) 13
per hour. If a gap is defined as the (c) 17 (d) 20
time difference between two

successive vehicle arrivals (with [GATE-2017-ME-SESSION-2]
vehicles assumed to be points), the 72. The standard deviation of linear

probability (up to four decimal dimensions P and Q are 3 m and

places) that the gap is greater than 8 4 m respectively. When assembled,

seconds is _________. the standard deviation (in m) of the
resulting linear dimension (P + Q) is
[GATE-2017-CE-SESSION-1] _______.

[GATE-2017-ME-SESSION-2]





173

Probability & Statistics

73. For a single server with Poisson 77. If a random variable X has a Poisson
arrival and exponential service time, distribution with mean 5, then the
the arrival rate is 12 per hour. Which expectation
one of the following service rates will
provide a steady state finite queue E   ( X + ) 2  2
length?   equals
_____________.
(a) 6 per hour (b) 10 per hour

(c) 12 per hour (d) 24 per hour [GATE-2017-PAPER-2-CS]


[GATE-2017-ME-SESSION-2] 78. The cumulative distribution function
of a random variable x is the
74. Let X be a Gaussian random variable probability that X takes the value
2
mean 0 and variance  . Let Y =
max (X, 0) where max (a, b) is the (a) less than or equal to x

maximum of a and b. The median of (b) equal to x
Y is _______________.
(c) greater than x
[GATE-2017]
(d) zero [ESE-2017-EC]
75. The marks obtained by a set of

students are 38, 84, 45, 70, 75, 60, 48. 79. For a random variable x having the
The mean and median marks, PDF shown in the figure given below
respectively, are

(a) 45 and 75 (b) 55 and 48


(c) 60 and 60 (d) 60 and 70

[GATE-2017 (CH)]

76. The following sequence of numbers

is arranged in increasing order: 1, x, The mean and the variance are,
x, x, y, y, 9, 16, 18. Given that the respectively
mean and median are equal, and are
also equal to twice the mode, the (a) 0.5 and 0.66 (b) 2.0 and 1.33
value of y is
(c) 1.0 and 0.66 (d) 1.0 and 1.33

(a) 5 (b) 6
[ESE-2017-EC]
(c) 7 (d) 8
80. The probability density function
[GATE-2017 (CH)] F ( ) x = ae − b x , where x is a random

variable whose allowable value range





174

Probability & Statistics

is from x = − to x = +. The CDF 83. What is the probability that at most 5
for this function for x  0 is defective fuses will be found in a box
of 200 fuses, if 2% of such fuses are
a defective?
bx
(a) e
b (a) 0.82 (b) 0.79

a (c) 0.59 (d) 0.82
−
(b) (2 e − bx )
b [ESE-2017-EE]

a 84. If X is a normal variate with mean 30
bx
(c) − be
b and standard deviation 5, what is
Probability (26  X  34 ) , given
a
−
+
(d) ( 2 e − bx ) [ESE-2017-EC] ( A z = 0.8 = ) 0.2881 where A
b
represents area.
81. A random variable X has the density (a) 0.2881 (b) 0.5762

1
function ( ) x = K , where (c) 0.8181 (d) 0.1616
f
+
1 x 2
− x  . Then the value of K is [ESE-2017-EE]

1 85. In a sample of 100 students, the mean
(a)  (b) of the marks (only integers) obtained
 by them in a test is 14 with its

standard deviation of 2.5 (marks
1
(c) 2 (d) obtained can be fitted with a normal
2 distribution). The percentage of

students scoring 16 marks is
[ESE-2017-EE]
(a) 36 (b) 23
82. A random variable X has a
probability density function (c) 12 (d) 10
 kx e x ; x  0
n −

f ( ) x =  (n is (Area under standard normal curve
 0; otherwise between z = 0 and z = 0.6 is 0.2257;

an integer) with mean 3. The value of and between z = 0 and z = 1.0 is
{k, n} are 0.3413) [ESE-2018-EE]

  1     1  
(a)  ,1 (b)  ,2 86. Consider a random variable to which
  2     4   a Poisson distribution is best fitted. It

  1   happens that P = 2 P on this
(c)  ,2 (d) {1, 2} (x= ) 1 3 (x= ) 2
  2   distribution plot. The variance of this


[ESE-2017-EE] distribution will be




175

Probability & Statistics

(a) 3 (b) 2 (c) Mean is greater than median and
mode
2
(c) 1 (d)
3 (d) Mode is greater than median

[ESE-2018-EE] [GATE-2018-CE (AFTERNOON
SESSION)]
86. The Graph of a function f(x) is shown
in the figure. 88. The arrival of customers over fixed
time intervals in a bank follow a
poisson distribution with an average
of 30 customers / hour. The
probability that the time between
successive customer arrival is
between 1 and 3 m minutes is
__________ (correct to two decimal
places).

For f(x) to be a valid probability [GATE-2018-ME (AFTERNOON

density function, the value of h is SESSION]

(a) 1/3 (b) 2/3
89. Let X and X bet wo independent
1 2
(c) 1 (d) 3 exponentially distributed random
variables with means 0.5 and 0.25,
[GATE-2018-CE (AFTERNOON respectively. Then Y-min ( X X )
,
SESSION)] 1 2
is
87. A probability distribution with right
skew is shown in the figure. (a) exponentially distributed with
mean 1/6

(b) exponentially distributed with
mean 2


(c) normally distributed with mean ¾

(d) normally distributed with mean
1/6
The correct statement for the
probability distribution is [GATE-2018-ME-AFTERNOON

SESSION]
(a) Mean is equal to mode

(b) Mean is greater than median but
less than mode




176

Probability & Statistics

90. A six-faced fair dice is rolled five 93. Type II error in hypothesis testing is
times. The probability (in %) of
obtaining “ONE” at least four times (a) acceptance of the null hypothesis
is when it is false and should be rejected

(b) rejection of the null hypothesis
(a) 33.3 (b) 3.33
when it is true and should be accepted
(c) 0.33 (d) 0.0033
(c) rejection of the null hypothesis

[GATE-2018-ME-MORNING when it is false and should be rejected
SESSION]
(d) acceptance of the null hypothesis
91. Let X , X be two independent when it is true and should be accepted
1
2
normal random variables with means [GATE-2016-SET-1]
 ,  and standard deviations  ,
1 2 1
 , respectively. Consider 94. Let X and Y are independent and
2
Y = X − X ; =  = 1, = 1, = 2 identically districted uniform random
1 2 1 2 1 2 variables over the interval (0, 1) and
. Then let S = X + Y. Find the probability

(a) Y is normally distributed with that the quadratic equation 9x-
mean 0 and variance 1 2 +9Sx+1=0 has no real root.

(b) Y is normally distributed with [MS 2005]

mean 0 and variance 5 95. Let E and F be two mutually disjoint

(c) Y has mean 0 and variance 5, but events. Further Let E and F be
is NOT normally distributed independent of G. If
( )
p = P E + ( ) P F and q = P(G),
(d) Y has mean 0 and variance 1, but then (E  
is NOT normally distributed P F G ) is _________

−
2
[GATE-2018-ME-MORNING (a) 1 pq (b) q + p
SESSION]
+
2
+
(c) p q (d) p q − pq
92. Let X , X , X and X be
3
2
1
4
independent normal random variables [MS 2006]
with zero mean and unit variance.
96. Let X be a continuous random
The probability that X is the
4 variable. With the probability density
smallest among the four is ________. function symmetric about O. If
( )   , then which of the
[GATE-2018-EC] V X
following statements is true?





177

Probability & Statistics

E x
( )
(a) ( ) = E X 99. Let X be a random variable having
probability density function
V x
(b) ( ) V X= ( )  ax  x 

0

)
f ( , ,x x  =  x + 1 x 0

0
( )
V x 
(c) ( ) V X  0 x  x 0


( )
V x 
(d) ( ) V X [MS 2006] where   0, x  . If y = ln     X    
0
0
x
97. An archer makes 10 independent , then (Y  ) 3 is _______  0 
P
attempts at a target and his
probability of hitting the target at (a) e − 3 x 0 (b) 1 e − 3 x 0


−
5
each attempt is . Then the
−
6 (c) e − 3 (d) 1 e − 3
conditional probability that his last
two attempts are successful given that [MS 2007]
he has a total of 7 successful attempts 100. Independent trails consisting of
is _______ rolling a fair die are performed. The
probability that 2 appears before 3
1 7
(a) (b) (or) 5 is __________
5 5 15
1 1
7
25 8!     3 (a) (b)
1
5
(c) (d)     2 3
    
  
36 3!5! 6 6
   
1 1
[MS 2006] (c) (d)
4 5
k x
f
x
98. Let ( ) x = 4 , −  . [MS 2007]
(1+ ) x
101. Let X X ,........X be independent
,
6
2
1
Then the value of k for which f(x) is a random variables such that
probability density function is
_______ P ( X = − = P ( X = ) 1 = 1 ,
) 1
i
i
2
1 1 i = 1,2,............6. Then
(a) (b)
6 2  6 

P    X = 4 is ________.

i
(c) 3 (d) 6  i= 1 
[MS 2006] (a) 3 (b) 3
32 4




178

Probability & Statistics

3 3 probability that the selected
(c) (d)
64 16 individual actually has the disease is
_______
[MS 2007]
(a) 0.01 (b) 0.05
102. Let E and F be two events such that
0  P ( ) 1E  and (c) 0.5 (d) 0.99


( / F +
=
( /
P E ) P E F C ) 1. Then [MS 2009]
____ 105. Let X be a random variable with
mean µ and variance 9. Then the
(a) E and F are mutually exclusive
smallest value of m such that

(b) E and F are independent P ( X −   m )  0.99 is
(
=
P
(c) ( E C / F + ) P E C / F C ) 1 __________
(a) 90 (b) 90
C
C
(d) P(E/F)+P(E /F )=1 [MS 2007]
100
103. Let X be Poisson (2) and Y be (c) (d) 30
Binomial (10,3/4) random variables. 11
If X and Y are independent then [MS 2009]

P ( XY = ) 0 is ________
106. The random variable X has the
 1  10 cumulative distributive Function
−
(a) e + − 2       (1 e − 2 )  0 if x  0
 4  

 1
 1  10  3 if x = 0


(b) e + − 2       (1 2e− − 2 ) F ( ) x = 
1 x
 4   + if 0 x  1


 3

1
10
(c) −2 ( )  1 if x  1

4 
 4  10 then E(X) equals ____________
(d) e + − 2 1−       [MS 2008]
1
 10  (a) (b) 1
3
104. For detecting a disease, a test gives
correct diagnosis with probability (c) 1 (d) 1
0.99. It is known that 1% of a 6 2

population suffers from this disease.
If randomly selected individual from [MS 2009]
this population test positive then the




179

Probability & Statistics

lnU up tails) = P (both coins show up
107. If Y = 1 , where U
lnU + 1 ln − (1 U 2 ) 1 heads) then u + v = _________

and U are independent U (0, 1) 1 1
2 (a) (b)
random variables, then variance of Y 4 2
= _________
3
(c) (d) 1
1 1 4
(a) (b)
12 3
[MS 2010]
1 1
(c) (d) 111. Let X be a discrete random variable
4 6 with

[MS 2009] 2 e − 1   1   K+ 2 2
P ( X = K ) =  +      ,
108. If X is a Binomial (30, 0.5) random 3 K !  3  3
variable, then _________ K = 0, 1, 2, ……., Let E = {0, 2, 4,
…..}. Then ( X  E ) =
P
P
(a) ( X  15 = ) 0.5 ___________

P
(b) ( X  15 = ) 0.5 5 2 5 1
1
−
−
1
(a) + e (b) + e
12 3 12 3
P
(c) ( X  15  ) 0.5
7 1 7 1
−
−
1
1
(c) − e (d) + e
P
(d) ( X  15  ) 0.5 [MS 2009] 12 3 12 3
109. Let E and F be two events with [MS 2010]
( ) 0, (F E =
P E  P / ) 0.3 and
112. If X and X are identical
1
2
=
P E
P ( E  F C ) 0.2 then ( ) = independent random variables N (0,
2
2
___________ 1), then ( X + X  ) 2 =______
P
1
2
1 2 − −
2
1
(a) (b) (a) e (b) e
7 7
−
1
−
(c) 1 e (d) 1 − −2
4 5
(c) (d)
7 7 [MS 2010]
[MS 2010] 113. Let X and X be identical
1
2
independent random variables Exp
110. Two coins with probability of heads u
P
and v respectively, are tossed (3). Then ( X + X  ) 1 = _______
1
2
independently. If P(both coins show

180

Probability & Statistics

−
−
3
3
(a) 2e (b) 3e 117. Let E and F be two events with P(E)
= 0.7, P(F) = 0.4 and
−
−
3
3
=
(c) 4e (d) 5e P ( E  F C ) 0.4 . Then
[MS 2011] P ( F E  / F C ) = __________
114. A four digit number is choosen at 1 1
random. The probability that there is (a) 2 (b)
3
exactly two zeroes in that number is 1 1
____________ (c) (d)
4 5

(a) 0.73 (b) 0.973 [MS 2013]

(c) 0.027 (d) 0.27 118. The probability that a hand of 5

playing cards contains at least two
[MS 2012]
aces is _________
115. A person makes repeated attempts to
4 C  48 C
destroy a target. Attempts are made (a) 2 3
independent of each other. The 52 C 5
probability of destroying the target in
any attempt is 0.8. Given that he fails (b) 4 C + 48 C 3
2
to destroy the target in first five 52 C 5
attempts, the probability that the 4 C  48 C + 48 C + 48 C 
th
target is destroyed in the 8 attempt is (c) 2   3 2 1 
52 C
(a) 0.128 (b) 0.032 5
4 C  48 C + 4 C  48 C + 4 C  48 C
(c) 0.160 (d) 0.064 (d) 2 3 3 52 C 5 2 4 1

[MS 2012]
[IISC 2003]
116. Let the random variable
)
X ~ B (5,P such that 119. Bag A contains 3 white and 4 red
balls and Bag B contains 6 white and
P ( X = ) 2 = 2P ( X = ) 3 . Then the 3 red balls. A biased coin, twice as
variance of X is ________ likely to come up heads as tail is
tossed once. If it shows head, a ball is
10 10
(a) (b) drawn from bag A. Otherwise from
3 9 bag B. Given that a white ball was

drawn, what is the probability that the
5 5
(c) (d) coin came up tail?
3 9
[JAM 2005]
[MS 2012]



181

Probability & Statistics

120. Six identical fair dice are thrown 123. Seven car accidents occurred in a
independently. Let S denote the week. What is the probability that
number of dice showing even
numbers on their upper faces. Then (i) They all occurred on the same day
the variance of the random variable S (ii) No two accidents occur on the

is _________ same day of week.

1 124. A die is rolled two times. Find the
(a) (b) 1
2 probability that


3 (i) Same face appears
(c) (d) 3
2
(ii) Sum is 10

[MS 2005]
(iii) Sum is greater than 10
121. Let X X 2 ,...........X be the (iv) Sum is neither 8 nor 9
,
21
1
random sample from a distribution
nd
1 21 (v) The 2 toss results in a value that
having variance 5. Let X =  X , is higher than first toss.
i
21 i= 1
21 2 125. Four fair six-sided dice are rolled.
S = (  X − X ) . Then the value of
i= 1 i The probability that the
E(S) = __________.
(i) Sum of the results is 22
(a) 5 (b) 100
(ii) Sum of the results is 21
(c) 0.25 (d) 105
(iii) Sum of the results is 20
122. In three independent throws of a fair

dice. Let X denote the number of 126. A man alternately tosses a coin and
upper faces showing six. Then the throws a die beginning with a coin.
What is the probability that he gets
2
value of (3E − X ) is __________ head before he gets 5 or 6 on the die?


20 2 127. Two cards are drawn at random from
(a) (b)
3 3 a pack of 52 cards. What is the
probability that
5 5
(c) (d) (i) both of them from same suit?
2 12

(ii) both of them from different suit?
[MS 2005]
128. A card is selected at random from a

pack of 52 cards. What is the
probability that it is



182

Probability & Statistics

(i) Spade (or) face card 133. Three unbiased dice of different
colours are rolled. The probability
(ii) King (or) red card
that the same number appears on

(iii) King (or) queen card atleast two of the three dice is

1
129. A & B stand in a ring with 10 other (a) 2 (b)
persons. If the arrangement of the 9 3
twelve persons is at random, find the

probability that there are exactly three (c) 4 (d) None
persons between A & B. 9

130. A number is selected at random from 134. An urn contains 5 red and 7 green
first 200 natural numbers. Find the balls. A ball is drawn at random and
probability that the number is its colour is noted. The ball is placed

divisible by 6 or 8. back into the urn along with another
ball of the same color. The
131. A point is selected at random inside a probability of getting a red ball in the
circle. Find the probability that the next draw is
point is nearer to the centre of the
circle than to its circumference. 65 67
(a) (b)
156 156
132. In a class of 100 students, 40 failed in
mathematics, 30 failed in physics, 25 79 89
failed in chemistry, 20 failed in maths (c) 156 (d) 156
and physics, 15 failed in physics &
chemistry, 10 failed in chemistry and Conditional Probability

Maths, 5 failed in Maths, physics and 135. A ticket is selected at random from
chemistry. If a students selected at 100 tickets numbered (00, 01, 02,
random, then find the probability that
….., 99). If X & Y denote the sum
(i) he passed in all three subjects and the product of the digits on the
tickets respectively, the value of P(X
(ii) he failed in atmost one subject = 9/ Y = 0) is

(iii) he failed in exactly one subject 1 2
(a) (b)
(iv) he failed in atleast two subjects 19 19


(v) he failed in atmost two subjects 3 4
(c) (d)
(vi) he failed in exactly two subjects 19 19








183

Probability & Statistics

136. If the probability that a 0.95 and the probability that a
communication system has a high transmitted ‘1’ is received as a ‘1’ is
fidelity is 0.81 and the probability 0.90. If the probability that a ‘0’ is
that it will have high fidelity and high transmitted is 0.4, find the probability
selectivity is 0.18. What is the that

probability that a system with high
fidelity will also have high (i) a ‘0’ is received
selectivity? (ii) a ‘1’ is received

2 3 (iii) a ‘1’ was transmitted given that a
(a) (b)
9 9 ‘1’ was received.

4 5 (iv) a ‘0’ was transmitted given that a
(c) (d)
9 9 ‘0’ was received.

137. An unbalanced die (with six faces, 140. The probability that a student knows

numbered from 1 to 6) is thrown. The the correct answer to a multiple
probability that the face value is odd choice question is 2 . If the student
is 90% of the probability that the face 3
value is even. The probability of does not know the answer, then the
getting an even numbered face is student guesses the answer. The
same. If the probability of face is probability of guessed answer being
even, given that it is greater than 3, is 1
0.75. What is the probability that the correct is 4 . Given that the student

face value exceeds 3? has answered the question correctly,
the conditional probability that the
Baye’s Theorem
student knows the correct answer is

138. The chances that doctor A diagnoses
a disease X correctly is 60%. The (a) 2 (b) 3
chances that a patient will die by his 3 4
treatment after correct diagnosis is 5 8
40% and the chances of death by (c) 6 (d)
9
wrong diagnosis is 70%. A patient of
doctor A who had disease X died. Random Variables
What is the probability that his

disease was diagnosed correctly? 141. A fair coin is tossed until a head or
four tails occur. The expected number
139. For a certain binary communication of tosses required is _______.
channel, the probability that a
transmitted ‘0’ is received as a ‘0’ is





184

Probability & Statistics

142. Rock bolts have length L = (150 + X) (a) 0.2 (b) 0.4
cm, where X is a random variable
with PDF. (c) 0.5 (d) 0.6

 1 Joint Distribution

−
2
x

f ( ) x =  4 (1 3x ) if −   2 146. The joint probability mass function of

 0, Otherwise (X, Y) is given below.

If 95% of the bolt lengths (L) lie in x→ 1 2 3
the interval 150 – C cm to 150 + C y↓ 3k 6k 9k
0
cm, the value of C is _______. 1 5k 8k 11k
2 7k 10k 13k
143. A random variable X has probability
density function as given below: Find (i) k

P
(ii) (x  ) 1

f ( ) x = + for 0 x  1
a bx

= 0 Otherwise (iii) (1 x  3, y  ) 1

P
2
E
If expected value ( ) x = , then 147. Two random variables X and Y are
3 distributed according to
P x  0.5  is
r

 x + 0   1,0  y 

144. A random variable X has the f ( , x y =  y x 1
)
following probability function. , x y  0 otherwise


X 0 1 2 3 4 5 6
P(X) K 3 5 7 9 11 13 The probability
K K K K K K
Find   1     1  
(i) p x  2    (ii) p y  2   



P
(i) K (ii) (3 x  ) 6    
 1 1 
(iii) Mean (iv) Variance (iii) p x    / y     
 2 2 
145. In a lottery there are 200 prizes of Rs.

5, 20 prizes of Rs. 25 and 5 prizes of (iv) ( p x + y  ) 1
Rs. 100. Assuming that 10,000 prizes
tickets are to be issued and sold.
What is the fair price to pay for the
ticket? (or if some one purchases a
lottery ticket his expectations is)






185

Probability & Statistics

Binomial Distribution 1 7
(a) 5 (b)
148. The probability of getting a total of 7 5 15
at least once in three tosses of a pair 25 8!     3
7
1
5
of fir dice is (c) (d)    
    
  
36 3!5! 6 6
   
125 91
(a) (b) 152. Out of 2000 families with 4 children
216 216
each. How many families would you
117 99 expect to have atleast one boy?
(c) (d)
216 216
(a) 1250 (b) 1875
149. The mean and variance of number of (c) 1500 (d) 1825
heads resulting from 10 independent
tosses of a fair coin respectively, are Poission Distribution


5 153. The second moment of a poisson –
(a) 5, (b) 10, 5
2 distributed random variable is 2. The
mean of the random variable is
5 5 1 1
(c) , (d) , 154. An observer counts 240 veh/h at
2 4 2 4
specific highway location. Assume

150. Consider an unbiased cubic die with that the vehicle arrival at the location
opposite faces coloured identically is poisson distributed. Find the
and each face coloured red, blue or probability of having
green such that each colour appears
only two times on the die. If the die is (i) One vehicle arriving over a 30
thrown thrice, the probability of second time interval.
obtaining red colour on the top face (ii) Atleast one vehicle arriving over
of the die atleast twice is ______ a 30 second time interval.


151. An archer makes 10 independent (iii) More than 2 vehicles arriving
attempts at a target and his over a 30 second time interval.
probability of hitting the target at
5 155. Let X be a binomial random variable
each attempt is . Then the 3
6 with n = 10 and P = and Y be
conditional probability that his last 4
two attempts are successful given that poission random variable with mean
he has a total of 7 successful attempts  = 2. If X, Y are independent, then

is P ( XY = ) 0 is







186

Probability & Statistics

 1  10 158. Among 10,000 random digits, find
−
(a) e + − 2       (1 e − 2 ) the probability that the digit 3 appears
 4 
atmost 950 times. (Area under normal
 1  10 curve between Z = 0 and Z = 1.67 is
−
(b) e + − 2       (1 2e − 2 ) 0.4525)
 4 
Exponential Distribution
 1  10
(c) e − 1       159. A continuous random variable X has
 4 
probability density function given by
 1  10

(d) e + − 2 1−        2e − 2x x  0
 4  f ( ) x =  0, otherwise


Normal (Gaussian) Distribution
The mean and variance of X are
156. For a random variable
1 1
1 1
)
x
X (−   following normal (a) , (b) ,
distribution, the mean is  = 100. If 2 8 2 4
=
the probability is P  for x  110. 1 1
Then the probability of x lying (c) 1, (d) 2,
2
2
between 90 & 110 i.e;
P (90  x  110 ) and equal to 160. If X is exponentially distributed, the
probability that X exceeds its
−
(a) 1 2 (b) 1 − expected value is _______

 161. The length of the shower on a tropical
(c) 1− (d) 2 island during rainy season has an
2
exponential distribution with
157. If the masses of 300 students are parameter 2, time being measured in
normally distributed with mean 68 minutes. What is the probability that
kgs and standard deviation 3 kgs. a shower will last more than 3 min?

How many students have masses
Uniform Distribution
(i) Greater than 72 kg
162. A random variable X is uniformly
(ii) Less than or equal to 64 kg distributed in the interval [0,1]. Find


(iii) between 65 & 71 kg (both (i) E(X) (ii) ( )
E X
2
inclusive)
3
E X
(iii) ( ) (iv) Variance
(Area under normal curve between Z
= 0 & Z = 1.33 is 0.4082)




187

Probability & Statistics

163. A point is randomly selected with (i) What is the mean of the
uniform probability in the XY-plane distribution?
with in the rectangle with corners at
(0,0), (1,0), (1,2) and (0,2). If P is the (a) 37.2 (b) 38.1
length of the position vector of the (c) 39.2 (d) 40.2

2
point, the expected value of p is
(ii) What is the median of the
2 distribution?
(a) (b) 1
3 (a) 37 (b) 38


4 5 (c) 39 (d) 40
(c) (d)
3 3
(iii) What is the mode of the
164. A passenger arrives at a bus stop at distribution?
10 AM, knowing that bus will arrive
at some time uniformly distributed (a) 38.33 (b) 40.66
between 10 AM and 10:30 AM (c) 42.66 (d) 43.33

167. The mean of five observations is 4
(i) What is the probability that he will
have to wait longer than 10 minutes. and their variance is 5.2. If the first
three values are 1, 2 and 6, then the
(ii) If at 10.15 AM the bus has not yet remaining two values are
arrived, what is the probability that he
will have to wait atleast 10 additional (a) 2 and 9 (b) 3 and 8

minutes. (c) 4 and 7 (d) 5 and 6

Statistics 168. The two lines of regression are
−
=
2x − − 20 0, 2y x + =
y
4 0
165. For the sample 27, 35, 40, 35, 36, 36,
29, find mean, median, mode and (i) The correlation coefficient is
standard deviation.
1 1
166. Consider the following frequency (a) 2 (b) −
2
distribution
1 1
Class Frequency (c) − (d)
0-10 4 4 4
10-20 5
20-30 7
30-40 10
40-50 12
50-60 8
60-70 4




188

Probability & Statistics
−
1 v = X Y what will be the
(ii) If  = , then what is  is
y x correlation coefficient between u &
4
equal to v?

1 (a) 0.433 (b) 0.333
(a) 1 (b)
2 (c) 0.233 (d) 0.133


1 171. The properties for a bivariate
(c) (d) 4
4 distribution of two random variables

X and Y are given below.
(iii) The means of x and y are
( ) 36,
( ) 24,
E X = E Y =
4
(a) x = 12, y = −
( ) 1524 ,
E X 2 = E Y 2 =
( ) 702
4
(b) x = − 12, y =
E ( XY =
) 1004
4
(c) x = − 12, y = −
The correlation coefficient between X
(d) x = 12, y = 4 and Y is

169. Correlation between two variables X 172. If byx = 0.7 and bxy = 0.2 then r =
and Y is given to be 0.6. These ___________.

variables are transformed to new
variables u = − 2X + 3 and Hypothesis Testing

v = 5Y − 2 173. To determine whether the mean
breaking strength of synthetic fibre
What will be the correlation
coefficient between u & v? produced by a certain company is 8
kg or not, a random sample of 50
(a) 0.6 fibres were tested yielding a mean
breaking strength of 7.8 kg with a
(b) -0.6
standard deviation of 0.5 kg. Test at

(c) 0.2 0.01 level of significance
(Z = 2.575 ) .
(d) Information is insufficient  /2

174. Can it be concluded that the average
170. Correlation between two variables X
1 life span of Indians is more than 70
and Y is given to be and their years if a random sample of 100
2 Indians has an average life span of
variances 36 & 16 respectively. 71.8 hours with a standard deviation
These variables are transformed to
the new variables u = X Y and
+



189

Probability & Statistics


of 8.9 years. Use 5% level of (d) ( A B = + +
P
) 1 p q
significance. z = z = 1.645 
 0.05
178. Let X be a random variable with the
175. The mean life time of a sample of 25 cumulative distribution function
bulbs is found as 1550 hours with a 
standard deviation of 120 hours. The  0, for x  0

1 x
company manufacturing the bulbs  + 2 for 0  x  1

claims that the average life of their  10


bulbs is 1600 hours. Is the claim F ( ) x =  3 x 2
+

acceptable at 5% loss?  10 for  1 x  2

t − 0.05 for 24 dof = 1.71 .  1, for x  2


1 1
176. If ( ) A = and ( ) = , then Which of the following statements is
P B
P
3 4 (are) TRUE?
which of the following is correct?
3

P
1 (a) (1 X  ) 2 =
)

(a)  P ( A B 10
3
3

P
1 (b) (1 X  ) 2 =
)
(b)  P ( A B 5

3
1
P 
) 2 =
(c) (1 X 


P
(c) ( A B  ) 1 2
12
4
(d) (1 X 
) 2 =
P 

P
(d) ( A B  ) 1 5
3
[JAM MS 2019]
177. The probability of the simultaneous
occurrence of two events A and B if 179. Let P be a probability function that
p. If the probability that exactly one assigns the same weight to each of
of A and B is q. Then which of the the points of the sample space
following is correct?  = 1,2,3,4. Consider the events
E =  , F =   and G =  
1,3
3,4
1,2
( ) ( )
(a) P A + P B = − − . Then which of the following
2 2p q
(
−
−
(b) P A B ) = 1 p q statement(s) is (are) true?
(a) E and F are independent
p (b) E and G are independent
P
(c) ( A B A B /  ) =
+
p q
(c) F and G are independent

190

Probability & Statistics

(d) E, F and G are independent 1
(a) c =
[JAM MS 2018] 8


180. Let S be the set of all 3 3 matrices (b) c = 8
having 3 entries equal to 1 and 6 (c) X and Y are independent
entries equal to 0. A matrix M is
picked uniformly at random from the (d) P (X = Y) = 0 [NET JUNE
set S. Then 2017]

:
1 183. Let X i   1 be a sequence of
i
(a) P { M is nonsingular } =
14 independent random variables each
having a normal distribution with
1
(b) P { M has rank 1 } = mean 2 and variance 5. Then which
14 of the following are true


1 1 n
(c) P { M is identity } = (a)  X converges in probability
14 i n 1 = i

1 to 2.
(d) P { trace (M) = 0 } =
14 1 n
2
(b)  X converges in probability
i
181. Suppose A, B, C are events in a i n 1 =
common probability space with P(A) to 9.

= 0.2, P(B) = 0.2, P(C) = 0.3,   2


P ( A B = ) 0.1, P ( A C = ) 0.1, (c)   1 i n n X i      converges in

P (B C = ) 0.1. Which of the  1 = 
probability to 4.
following are possible values of
P ( A B C  )? n   X i   2
(d)      converges in
i= 1  n 
(a) 0.5 (b) 0.3
probability to 0.
(c) 0.4 (d) 0.9
[NET DEC 2016]

182. Let c R be a constant. Let X, Y be 184. A and B are two events defined as
random variables with joint follows:
probability density function
A: It rains today with P(A) = 40%
 cxy for 0 x   1

y

)
f ( ,x y =  .
 0 otherwise B: It rains tomorrow with P(B) = 50%

Which of the following statements Also, P(it rains today and tomorrow)
are correct? = 30%


191

Probability & Statistics
( (
))


:
Also, E P A B ) ( / A B and 187. If A and B are two independent
1
events such that P(A) = 1/2, P(B) =
 (
 (


E 2 : P A B or B  A ) ( / A B )) 1/5, then
P
) 1/2
. Then which of the following is / are (a) ( / A B =
true?
 A  5
(a) A and B are independent (b) P       =

 A B  6
(b) P(A/B) < P(B/A)
 A B 

(c) P      = 0
(c) E and E are equiprobable  ' A B '  

1
2
(
(d) ( /P A A B )) = P ( B ( / A B )) (d) None of these
185. Two numbers are chosen from {1, 2, 188. If A and B are two independent
(

3, 4, 5, 6, 7, 8} one after another events such that P A B = ) 2/15
without replacement. Then the ( )

probability that and P A B = 1/6, then P(B) is
(a) the smaller value of two is less (a) 1/5 (b) 1/6

than 3 is 13/28
(c) 4/5 (d) 5/6
(b) the bigger value of two is more
than 5 is 9/14 189. The probability that a 50-year-old
man will be alive at 60 is 0.83 and the
(c) product of two number is even is probability that a 45 year old

11/14 woman will be alive at 55 is 0.87.
then
(d) none of these
(a) the probability that both will be
186. P(A) = 3/8; P(B) = 1/2; alive is 0.7221


P ( A B = ) 5/8, which of the
following do / does hold good? (b) at least one of them will alive is
0.9779
) 2P A B
P
(a) ( A C / B = ( / C )
(c) at least one of them will alive is
)
(b) ( ) = P ( / A B 0.8230
P B
(d) the probability that both will be
(
(c) 15P A C / B C ) 8P= ( /B A C ) alive is 0.6320



(d) ( /P A B C ) ( A B=  )





192

Probability & Statistics

190. Two busses A and B are scheduled to 27 18
arrive at a town central bus station at (c) 75 (d) 25
noon. The probability that bus A will 193. A box contains 5 black balls and 3
be late is 1/5. The probability that bus red balls. A total of three balls are
B will be late is 7/25. The probability picked from the box one after

that the bus B is late given that bus A another, without replacing them back.
is late is 9/10. Then, The probability of getting two black
balls and one red ball is
3
(a) probability that neither bus will be (a) (b) 2
late on a particular day is 7/10 8 15

(b) probability that bus A is late given 15 1
that bus B is late is 18/28 (c) 28 (d)
2
[GATE-1997]
(c) probability that at least one bus is 194. Seven car accidents occurred in a
late is 3/10 week, what is the probability that


(d) probability that at least one bus is they all occurred on the same day?
in time is 4/5 (a) 1 (b) 1
7 7 7 6
191. The chance of a student passing an

exam is 20%. The chance of a student (c) 1 (d) 2
passing the exam and getting above 2 7 2 7
90% in it is 5%. Given that a student [GATE-2001]
passes the examination, the
probability that the student gets above 195. Manish has to travel from A to D
90% marks is changing buses at stops B and C
enroute. The maximum waiting time

1 1 at either stop can be 8 minutes each,
(a) (b) but any time of waiting up to 8
18 4 minutes is equally likely at both

places. He can afford up to 13
2 5
(c) (d) minutes of total waiting time if he is
9 18 to arrive at D on time. What is the
probability that Manish will arrive
[GATE 1992] late at D?
192. The probability that a number 8 13
selected at random between 100 and (a) (b)
999 (both inclusive) will not contain 13 64
the digit 7 is 119 9

16  9  3 (c) (d)
(a) (b)       128 128
25  10  [GATE-2002-ME]







193

Probability & Statistics

196. In class of 200 students, 125 students 199. The box 1 contains chips numbered 3,
have taken programming language 6, 9, 12 and 15. The box 2 contains
course, 85 students have taken data chips numbered 11, 6, 16, 21 and 26.
structures course, 65 students have Two chips, one from each box are
taken computer organization course, drawn at random.
50 students have taken both The number written on these chips
programming languages and data are multiplied. The probability for the
structures, 35 students have taken product to be an even number is
both programming languages and _____
computer organization, 30 students 6 2
have taken both data structures and (a) 25 (b)
5
computer organization, 15 students
have taken all the three courses. How 3 19
many students have not taken any of (c) (d)
the three courses? 5 25
(a) 15 (b) 20 [GATE 2009 (IN)]
200. A and B friends. They decide to meet
(c) 25 (d) 35 between 1PM and 2 PM on a given
[GATE-2004 (IT)] day. There is a condition that
197. A bag contains 10 blue marbles, 20 whoever arrives first will not wait for
black marbles and 30 red marbles. A the other for more than 15 minutes.
marble is drawn from the bag, its The probability that they will met on
colour recorded and it is put back in that day is
the bag. This process is repeated 3 (a) 1 (b) 1
times. The probability that no two of 4 16
the marbles drawn have the same
colour is 7 9
1 1 (c) (d)
(a) (b) 16 16
36 6 201. Assume for simplicity that N people,
all born in April (a month of 30
1 1 days), are collected in a room.
(c) (d)
4 3 Consider the event of at least two
[GATE-2005 (IT)] people in the room being born on the
198. In a game, two players X and Y are same date of the month, even if in
tossing a coin alternately. Whoever different year, e.g. 1980 and 1985.
gets a ‘head’ first, wins the game and What is the smallest N so that the
the game is terminated. Find the probability of this event exceeds 0.5?
chance that player X will win the (a) 20 (b) 7
game if he starts?
(a) 1/3 (b) 1/2 (c) 15 (d) 16
[GATE-2009-EE]
(c) 2/3 (d) 3/4







194

Probability & Statistics

202. What is the probability that a divisor probabilities of failure of the
96
99
of 10 is a multiple of 10 ? machines are given as:
(a) 1/625 (b) 4/625 P = A 0.15, P = B 0.05 and P = 0.1
C
(c) 12/625 (d) 16/625
[GATE-2010 (CS)]
203. The probability that a student knows
the correct answer to a multiple
choice question is 2/3. If the student Assuming, independence of failures
does not know the answer, then the of the machines, the probability that a
student guesses the answer. The given job is successfully processed
probability of the guessed answer (up to the third decimal place) is
being correct is 1/4. Given that the [GATE-2014 (IN-SET1)]
student has answered the question 206. Parcels from sender S receiver R pass
correctly, the conditional probability sequentially through two post-offices.
that the student knows the correct 1
answer is Each post-office has a probability
2 3 5
(a) (b) of losing an incoming parcel,
3 4 independently of all other parcels.
5 8 Given that a parcel is lost, the
(c) (d)
6 9 probability that it was lost by the
[GATE-2013-ME] second post-office is ____
204. An automobile plant contracted to [GATE-2014-EC-SET 4]
buy shock absorbers from two 207. The security system at an IT office is
suppliers X and Y. X supplies 60% composed of 10 computers of which
and Y supplies 40% of the shock exactly four are working. To check
absorbers. All shock absorbers are whether the system is functional, the
subjected to a quality test. The ones officials inspect four of the computers
that pass the quality test are picked at random (without
considered reliable. Of X’s shock replacement). The system is deemed
absorbers, 96% are reliable. Of Y’s functional if at least three of the four
shock absorbers, 72% are reliable. computers inspected are working. Let
The probability that a randomly the probability that the system is
chosen shock absorber, which is deemed functional be denoted by p.
found to be reliable, is made by Y is Then 100p = _________.
(a) 0.288 (b) 0.334 [GATE-2014 (CS-SET2)]
(c) 0.667 (d) 0.720 208. The chance of a student passing an
[GATE-2012] exam is 20%. The chance of student
205. The figure shown the schematic of a passing the exam and getting above
production process with machines A, 90% marks in it is 5%. Given that a
B and C. an input job needs to be student passes the examination, the
preprocessed either by A or by B probability that the student gets above
before it is fed to C, from which the 90% marks is
final finished product comes out. The






195

Probability & Statistics

1 1 325
(a) (b) (a) 0 (b)
18 4 864
2 5 525 5
(c) (d) (c) (d)
9 18 864 12
[GATE-2015 (ME-SET2)] [GATE 2018 (EE)]
209. Candidates were asked to come to an 212. A person moving through a
interview with 3 pens each. Black, tuberculosis prone zone has a 50%
Blue, green and red were the probability of becoming infected.
permitted pen colours that the However, only 30% of infected
candidate could bring. The people developed the disease. What
probability that a candidate comes percentage of people developed the
with all 3 pens having the same disease. What percentage of people
colour is ____. moving through a tuberculosis prone
[GATE-2016-EE-SET 1] zone remains infected but does not
210. P and Q are considering to apply for a show symptoms of disease?
job. The probability that P applies for (a) 15 (b) 33
1 (c) 35 (d) 37
the job is . The probability that P [GATE-2016]
4
applies for the job given that Q 213. A pair of dice is rolled again and
1 again till a total of 5 or 7 is obtained.
applies for the job is , and the The chance that a total 5 comes
2 before a total of 7 is
probability that Q applies for the job 2 3
1 (a) (b)
given that P applies for the job is . 5 7
3 3
Then the probability that P does not (c) (d) none of these
apply for the job given that Q does 13
not apply for the job is 214. A bag P contains 3 white and 4 black
4 5 balls and another bag Q contains 4
(a) (b) white and three black balls. A ball is
5 6 transferred (at random) from bag P to
7 11
(c) (d) the bag Q and then a ball is
8 12 transferred from bag Q to the bag P.
[GATE-2017 PAPER-2 (CS)] A ball is then taken out from the bag
211. A class of twelve children has two P. The chance that it is a white ball is
more boys, then girls. A group of 31 25
three children are randomly picked (a) 56 (b) 49
from this class to accompany the 25
teacher on a field trip. What is the (c) (d) none of these
probability that the group 56
accompanying the teacher contains 215. There are two identical locks with
more girls than boy? two identical keys and the key are
among six different ones which a
person carries in his pocket. In hurry
he drops one key somewhere. Then



196

Probability & Statistics

the probability that the locks can still 1
opened by drawing one key at (c) 18 (d) none of these
random is equal to RANDOM VARIABLE
1 5
(a) (b) 220. Let X and Y be two independent
3 6 random variables. Which one of the
1 1 relations between expectation ©,
(c) (d) variance (Var) and covariance (Cov)
12 30
[GATE] given below is FALSE?
)
E
( ) ( )
216. A party of n persons takes their seats (a) ( XY = E X E Y
at random at a round table, then the (b) Cov X ) 0
( ,Y =
probability that two specified person © Var X + Y = ( ) Var Y
(
( )
) Var X +
do not sit together is
2
2 n − 3 (d) ( X Y 2 ) (E X 2 ( ))
=
2
( )) (E Y
E
(a) (b)
n − 1 n − 1 [GATE-2007-ME]
n − 2 1 221. If the standard deviation of the spot
(c) (d)
n − 1 n − 1 speed of vehicles in a highway is
[GATE] 8.8km/h and the mean speed of the
217. The letters of the word vehicles is 33 km/h, the coefficient of
PROBABILITY are arranged in all variation in speed is
possible ways. The chance that B’s (a) 0.1517 (b) 0.1867
and also two I’s occur together is © 0.2666 (d) 0.3646
1 2 [GATE-2007-CE]
(a) (b) 222. If the difference between the
55 55 expectation of the square of a random
4  2 

(c) (d) none of these variable ( ) E X and the square of
165  
218. From 6 positive and 8 negative the expectation of the random

( )
numbers 4 numbers are drawn at variable E X    2 is denoted by R,


random without replacement and then
multiplied, the probability that the (a) R = 0 (b) R < 0
product is a positive number is © R  0 (d) R > 0
505 50
(a) (b) [GATE-2011 (CS)]
1001 1001 223. A simple random sample of 100
5 55 observations was taken from a large
(c) (d)
101 1001 population. The sample mean & the
[GATE] standard deviation were determined
219. If two squares are chosen at random to be 80 to 12 respectively. The
on a chess board to probability that standards error of mean is ______
they have a side in common is [GATE-2014 (PI-SET1)]
1 2
(a) (b)
9 7







197

Probability & Statistics

224. Marks obtained by 100 students in an 227. The standard deviation of liner

examination are given in the table: dimensions P and Q are 3 m and
4 m respectively. When assembled,

Marks Number of
S.No the standard deviation (in m
obtained students ) of the
1 25 20 resulting linear dimension (P+Q) is
2 30 20 _____.
3 35 40
4 40 20 [GATE-2017 ME SESSION-II]
What would be the mean, median and
mode of the marks obtained by the 228. The following sequence of numbers
students? is arranged in increasing order : 1, x,
(a) Mean 33; Median 35; Mode 40 x, x, y, y, 9, 16, 18. Given that the
mean and median are equal, and are
(b) Mean 35; Median 32; Mode 40
also equal to twice the mode, the
(c) Mean 33; Median 35; Mode 35 value of y is


(d) Mean 35; Median 32; Mode 35 (a) 5 (b) 6

[GATE-2014 (PI-SET 1)] (c) 7 (d) 8


225. The spot speed (expressed in km/hr) [GATE-2017 (CH)]
observed at a road section are 66, 62,
45, 79, 32, 51, 56, 60, 53 and 49. The DISCRETE RANDOM VARIABLE
median speed expressed in km/hr is 229. The mean of squares of first 23
_____. natural number is ______.


[Note answer with one decimal 230. A random variate has the following
accuracy] distribution:

[GATE-2016 (CE-SET 2)] x : 0 1 2 3 4 5 6 7

2
p x 2k 2k 3k k 2 2k 2 7k + k
( ) : 0 k
226. A sample of 15 data is as follows 17,
18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9 20,
17, 3. The mode of the data is The value of k is _____.

(a) 4 (b) 13 231. A six-face fair dice is rolled a large

number of times. The mean value of
(c) 17 (d) 20
the outcomes is ____.
[GATE 2017 ME SESSION-II]
[GATE 2017]









198