Math 10

Probability of an event

Let S be the sample space of a random experiment and E the event which is a subset of the
sample space, then probability of happening an event E is denoted by P(E). Symbolically, it is
defined as,

P (E) = n(E)
n(S)

Where, n (E) = total number of favorable cases for event E,

n(S) = total number of all possible outcomes of equally likely cases

P (E) = probability of not getting an event E = 1– P(E)

Nature of probability

(i) The value of probability always lies between 0 and 1, i.e., 0 ≤ P(E) ≤ 1
(ii) If E is an impossible event, then P(E) = 0.
(iii) If E is a sure event, then P(E) = 1

Playing Cards:

♣◊♥ ♠
Club (13 cards) Diamond (13 cards) Heart (13 cards)
Spade (13 cards)

These are known as 4 suits of playing cards.

Red Cards ⇒ All the cards of Diamond and Heart (26 cards)

Black Cards ⇒ All the cards of Club and Spade (26 cards)

Oasis School Mathematics-10 345

Jack Queen King Ace

Face cards ⇒ Jack, Queen and King

There are 12 face cards in a pack of playing cards.
Mutually exclusive events:
Two or more events are said to be mutually exclusive events, if both of them cannot occur
together in the same trial. If A and B are two mutually exclusive events, then n(A∩B) = 0 and
P (A∩B) = 0.

Example : If a card is drawn from 52 cards,
(i) getting a king and an ace is a mutually exclusive event.
(ii) getting a diamond and a heart is a mutually exclusive event.

Additive law for mutually exclusive events:
If 'A' and 'B' are mutually exclusive events, then the probability of occurrence of at least one of
the events A and B is P (A∪B) = P(A) + P(B).

Mutually non-exclusive events:

Two events 'A' and 'B' are said to be mutually non-exclusive if both of them can occur in the

same trial. For example, if a card is drawn from a pack of 52 cards, the probability of getting a

face card and a red card is a mutually non-exclusive event.

Similarly, getting a king and a diamond is a mutually non-exclusive event.

In this type of event, P(A∪B) = P(A) + P(B)–P(A∩B)

P(A∪B) = 1–P (A∪B)

Worked Out Examples

Example: 1

'A' and 'B' are mutually exclusive events if P(A) = 1 , P(B) = 2 . Find (i) P(A∪B) (ii) P(A∩B).
5 3

Solution:

Here, given P(A) = 1 , P(B) = 2
5 3

346 Oasis School Mathematics-10

Since 'A' and 'B' are two mutually exclusive events, then we have

P(A∪B) = P(A) + P(B)

= 1 + 2
5 3
13
= 3+10 = 15
15

Again, P(A∪B) = 1–P (A∪B)

= 1– 13 = 125
15

Example: 2

A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of the
the card is an ace or a queen.

Solution:

Let 'A' and 'Q' be the event of getting an ace or a queen respectively.

Number of possible cases n(S) = 52

Then, Probability of getting an ace P(A) = n(A) = 4 = 1
n(S) 52 13

Probability of getting a queen P(Q) n(Q)
= n(S)

= 41

=

52 13

P (A or Q) = P(A) + P(Q)

= 1 + 1 = 123
13 13
∴ The probability that the card is an ace or a queen is 2 .
13

Example: 3

From a number of cards numbered 1 to 20, a card is drawn randomly. Find the probability
of getting a card whose number is divisible by 3 or 7.

Solution:

Here, Number of possible cases n(S) = 20

Let A be a set of numbers divisible by 3, then,

A = {3, 6, 9, 12, 15, 18} then, n(A) = 6

Let B be a set of numbers which are divisible by 7

i.e. B = {7, 14} then, n(B) = 2

Now, finding the possibility of event A

P(A) = n(A) = 6 = 3
Probability of event B n(S) 20 10

P(B) = n(B) = 2 = 1
n(S) 20 10

Oasis School Mathematics-10 347

Since, A and B are mutually exclusive events,

P(A ∪ B) = P(A) + P(B)

( )=3 + 1 = 4 = 2
10 10 10 5

∴ The probability of getting a card whose number is divisible by 3 or 7 having an ace or

queen is 2 .
5
Example: 4

From a pack of 52 cards, a card is chosen at random. Find the probability of getting a red
card or a face card.

Solution: Here, n(S) = 52

Let, 'A' and 'B' be the events of getting a red card and a face card respectively.

Then, nn(A(A) ) == 2626

nn(B(B) ) == 1212

nn(A(A∩∩BB) ) == 66

WWe ehhavave,e, PP(A(A) ) == nnn(n(AS((A)S)))==25622562

PP(B(B) ) == nnnn((BS((B)S) ))==15221522
PP(A(A∩∩BB) ) == nn(An(An(∩S(∩)SB)B) =) =562562
Since, ASiSanincedc,eBA, AaarnaendmdBBuatraueraemlmluyutnatouanull-lyelyxncnolnouns−iv−eexecxelvculeusnisvtisve,eevevenentst,s,

WWe ehhavave ,e ,

PP(A(A∪∪BB) =) =PP(A(A) +) +PP(B(B) −) −PP(A(A∩∩BB) )

==52262526++15221522−−562562
==35225322
==183183
p∴ro∴bPaP(Rb(Rieliedtydoroofrfafcgaeecetctacinradgrd)a)=r=e18d3183c. a. rd
∴ The or a face card is 8.
13
Example: 5

From a number of cards numbered from 1 to 21, a card is drawn at random. Find the

probability of getting a card whose number is divisible by 3 or 7.

Solution:
Here, Number of possible cases n(S) = 21

Let A be a set of numbers which are divisible by 3

i.e. A = {3, 6, 9, 12, 15, 18, 21} then, n(A) = 7
Let B be a set of numbers which are divisible by 7, i.e.,

B = {7, 14, 21} then, n(B) = 3

Also, A ∩ B = {21}

348 Oasis School Mathematics-10

i.e. n(A∩B) = 1
Now,

Probability of event A, P(A) = n(A) = 7 = 1
n(S) 21 3

Probability of event B, P(B) = n(B) = 3 = 1
n(S) 21 7

We have, P(A∪B) = P(A) + P(B) – P(A∩B)


= 1 + 1 –1
3 7 21

= 9
21
∴ The probability of getting a card whose number is divisible by 3 or 7 is 3 .
7



Exercise 18.1

1. Identify whether the given events are mutually exclusive or mutually non-exclusive.

(a) getting a king or an ace from a pack of 52 cards.

(b) getting a heart or a diamond from a pack of 52 cards.

(c) getting a heart or a king from the pack of 52 cards.

(d) getting a face card or a red card from a pack of 52 cards.

(e) getting a number which is a multiple of 5 or 7 from a stack of cards numbered 1
to 30.

(f) getting a number which is a prime number or an even number from a stack of
cards numbered 1 to 20.

(g) getting a number which is the multiple of 3 or 4 from a stack of cards numbered
2. (a)
1 to 25.
(b)
If 'A' and 'B' are two mutually exclusive events, where P(A) = 1 and P(B) = 13, find
(c) 2
P(A∪B).

Two events M and N are mutually exclusive events with P(M) = 1 and P(N) = 5 .
8 8
Find;

(i) P(M∪N) (ii) P(M∪N )

P(A∪B) = 3 and P(A) = 12. Find the value of P(B), if A and B are two mutually
4
exclusive events.

Oasis School Mathematics-10 349

3. (a) A marble is drawn from a box containing 15 black, 5 green, 10 red and 10 yellow
marbles. Find the probability of getting,

(i) A black marble or green marble.

(b) (ii) A green or a red marble.

(iii) A yellow or black marble.

(c) A card is drawn from a well shuffled pack of 52 cards. Find the probability of

getting,

(i) an ace or a king (ii) a black ace or a red king.

(d) (iii) a red jack or a queen. (iv) a face card or an ace.

(v) a heart or a spade (vi) a diamond or a black card.

4. (a) (vii) neither a king nor an ace.

(b) A card is drawn at random from a stack of cards numbered 5 to 30. Find the
probability of getting a
(c) (i) square numbered or cube numbered card.
(ii) multiple of 5 or 9.
5. (a) (iii) even numbered or prime numbered.
(iv) multiple of 5 or multiple of 7.

From a set of 15 cards numbered 1, 2, 3, 4 ……. 15, one is drawn at random. What
is the probability of getting card which is:
(b)
(i) divisible by 3 or divisible by 7 (ii) divisible neither by 3 nor by 7

(iii) multiple of 5 or 6

If 'A' and 'B' are mutually non-exclusive events and P(A) = 2 , P(B) = 6 and
P(A∩B) = 1 , find the value of (i) P(A∪B), (ii) P(A∪B). 5 25

25

If P(A) = 1 , P(B) = 1 and P(A∪B) = 3 , find the value of P(A∩B) if A and B are
3 7 7
mutually non-exclusive events.
1
If 'A' and 'B' are mutually non-exclusive events with P(A) = 2 , P(B) = 1 , and
. 4
5 (i)
P(A∪B) = 8 , find P(A∪B) and (ii) P(A∩B).

A card is drawn at random from a well shuffled pack of 52 cards. Find the

probability of getting,

(i) an ace or a red card. (ii) a red card or a face card.
(iii) an ace or a heart. (iv) a spade or a queen.

(v) a king or a red card (vi) a red queen or a heart.

From a pack of 52 cards, a card is drawn at random. Find the probability of
getting a black or a non-faced card.

350 Oasis School Mathematics-10

(c) Acard is drawn at random from a set of cards numbered 1 to 20. Find the probability
of getting prime numbered or an even numbered card.

(d) A card is drawn at random from a set of cards numbered 1 to 36. Find the
probability of getting a card of a multiple of 5 or 7.

(e) A card is drawn at random from a group of cards, numbered from 2 to 13. Find
the probability of getting a prime numbered or even numbered card.



Answer

1. Consult your teacher

2. (a) 5 (b) (i) 3 (ii) 1 (c) 1 3.(a) 1 (ii) 3 (iii) 5 (b)(i) 2 (ii) 1 (iii) 3 (iv) 4 (v) 1
6 4 4 4 2 8 8 13 13 26 13 2

(vi) 3 (vii) 11 (c)(i) 5 (ii) 9 (iii) 21 (iv) 5 (d)(i) 7 (ii) 8 (iii) 1 4.a.(i) 3 (ii) 2 b. 1
4 13 26 26 26 13 15 15 3 5 5 21

(c)(i) 3 (ii) 1 5.(a)(i) 7 (ii) 8 (iii) 4 (iv) 4 (v) 7 (vi) 7 (b) 23 (c) 17 (d) 11 (e) 11
8 8 13 13 13 13 13 26 26 20 36 12



18.3 Independent Events

Two or more events are said to be independent when the result of one event doesn't
affect the result of the other. For example, when a coin is tossed twice the result of the
first doesn't affect the result of the second event. So, they are called independent events.

Multiplicative law of probability

The probability of two or more independent events occurring together or in succession
is the product of their individual probabilities. Symbolically, if A and B are two
independent events then the probability of events A and B occurring together or in
succession is denoted by,

P (A and B) or P (AB) or P(A∩B)

Thus, it is written as

P(A and B) = P(A) × P(B)
P(A) × P(B),
or, P(A ∩ B) =

Where, P(A and B) = the probability of occurring two events A and B together or in
succession.

P(A) = Probability of occurring event A

P(B) = Probability of occurring an event B
If A, B and C are three independent events corresponding to a random experiment, then

P(ABC) = P(A) × P(B) × P(C)

i.e. P(A∩B∩C) = P(A) × P(B) × P(C)

Oasis School Mathematics-10 351

Worked Out Examples

Example: 1

Find the probability of getting a tail on a coin and 4 on a dice when a coin is tossed and dice
is rolled simultaneously.
Solution,

Let 'T' be the event of getting a tail on a coin and F be the event of getting a '4' on a dice.

When a coin is tossed, the sample space is

S = {H,T}, ∴ n (S) = 2

And the number of favorable cases n(E) = 1

Then probability of getting a tail P(T) = n(E) = 1
n(S) 2

Again, when a dice is thrown at once, the sample space is

S = {1, 2, 3, 4, 5, 6}, ∴ n(S) = 6

Number of favorable cases n(E) = 1

Then, probability of getting a 4, n(E) 1
n(S) 6
P (F) = =

Since these two events are independent events,

∴ P(T and F) = P(T) × P(F)

= 1 × 1 = 1
2 6 12
Example: 2

The probability of solving a problem by A is 2 and the probability of solving the problem
3
3
by B is 4 . Find the probability of

(i) solving the problem by both of them (ii) the problem being solved, if they both try.

Solution:

Here, we have 2
3
P(A) = 3
P(B) = 4


P(A and B) = P(A∩B) = ?

P (A or B) = P (A∪B) = ?

We have, P (A∩B) = P(A) × P(B) [∵ A and B are independent events]

= 2 × 3
3 4

= 1
2

352 Oasis School Mathematics-10

Now, using relation, P(A∪B) = P(A) + P(B) – P(A∩B)

= 2 + 3 – 1
3 4 2

= 8+9–6 = 11
12 12

Example: 3

A dice is thrown thrice. What is the probability that it will turn up,

(i) six in each time? (ii) no sixes? (iii) at least one 6?
Solution:

Here the possible cases for each throw

n(S) = 6

(i) Probability of getting a 6 each time is

P(6,6,6) = 11 1 = 1
6×6 ×6 216

(ii) Probability of not getting 6 = 1 – 1 = 5
66

P(no sixes) = 5 × 5 5 = 125
6 6 ×6 216

(iii) At least one six

Since the probability of not getting a 6 is P (no sixes) = 125
216

Then probability of getting at least one 6 is

P (at least one 6) = 1– 125 = 216-125 = 91
Example: 4 216 216 216

A card is drawn randomly from a pack of cards and a dice is thrown once. Determine the
probability of not getting a king on the card as well as 6 on the dice.

Solution:

Here, in a pack of cards n(S) = 52

Probability of getting a king, P(K) = 4 1=13113=
52
Probability of not getting a king, P(not K) = 1– 12
13

Again, in a throw of dice, n(S) = 6
1
Probability of getting 6, P(6) =
6
1
The probability of not getting 6, P (not 6) = 1– 6

= 5
Finally, probability of not getting a king as well as 6 =
P (not K and not 6) 6

P (not K) × P (not 6)

= 12 5 = 10
13 × 6 13

Oasis School Mathematics-10 353

Exercise 18.2

1. (a) If P(A) = 1 , P(B) = 1
3 4 , find P(A∩B) if 'A' and 'B' are independent events.

(b) If P(A) = 1 and P(A and B) = 1 , find P(B) if 'A' and 'B' are independent events.
3 8

2. (a) Find the probability of getting a 3 on a dice and head on a coin when the dice is
rolled and the coin is tossed simultaneously.

(b) A card is drawn randomly from a pack of 52 cards and a dice is thrown once.
Determine the probability of not getting a king as well as not getting 6 on the dice.

(c) A card is drawn at random from a pack of 52 cards, and at the same time a marble
is drawn at random from a bag containing 2 red marbles and 3 blue marbles. Find
the probability of getting:

(i) a king and a blue marble. (ii) a queen and a red marble.

(iii) a black card and a red marble. (iv) a face card and a blue marble.

3. (a) A and B appear in an interview for two vacancies of the same post. The probability

of A's selection is 1 and that of B's selection is 1 . What is the probability that
75

(b) (i) both of them will be selected?

(ii) at least one of them will be selected?

(iii) none of them will be selected?

The probability of solving a mathematical problem by two students 'A' and 'B' are 1
3
1
and 4 respectively. If the problem is given to both students, find the probability of;

(i) solving the problem by both of them.

(ii) solving the problem by at least one of them.

(iii) solving the problem by none of them.

4. (a) A coin is tossed and a dice is rolled together. Find the probability of getting.

(i) a head on the coin and 4 on the dice.

(ii) a tail on the coin and an odd number on the dice.

(b) A bag contains 4 red and 7 blue balls. A ball is drawn and replaced. If another ball
is drawn, find the probability of getting

(i) two red balls. (ii) two blue balls.

(c) (iii) blue in the first draw and red in the second draw.

(iv) a blue and a red ball.

A bag contains 6 blue, 5 white and 4 black balls. A ball is drawn randomly from
the bag. At the same time, a card is also drawn from a pack of 52 cards. Find the
probability of getting

(i) a white ball and a jack. (ii) a black ball and a black card.

354 Oasis School Mathematics-10

(d) A ball is drawn randomly from a bag containing 5 blue, 3 red and 2 white balls and
replaced. If a second ball is drawn, what is the probability of getting

(i) a blue ball in the first draw and a white ball in the second draw?

(ii) a red or white ball in the first draw and a blue ball in the second draw?

(iii) a blue ball in both draws or a white ball in both the draws?

(iv) both balls of the same color?

Answer 

1. (a) 1 (b) 3 2. (a) 1 (b) 10 (c) (i) 3 (ii) 2 (iii) 1 (iv) 9
12 8 12 13 65 65 5 65

3. (a)(i) 1 (ii) 11 (iii) 24 (b) (i) 1 (ii) 1 (iii) 1 4.(a)(i) 1 (ii) 1
35 35 35 12 2 2 12 4

(b)(i) 16 (ii) 49 (iii) 28 (iv) 28 (c)(i) 1 (ii) 2 (d)(i) 1 (ii) 1 (iii) 29 (iv) 19
121 121 121 121 39 15 10 4 100 50

18.4 Tree diagram

If we have to find the combined result of two or more events, we cannot always use a table to
show all possible outcomes, so in such a case, a probability tree diagram is suitable method to

show all possible outcomes.

The probability tree diagram is a diagrammatical representation of all the possible outcomes
in any type of random experiment. Generally, we use a tree diagram to find the probability of
any combined result of two or more arrows for each event starting from a point. Again, we use
the same way to represent the other event. Now, the possible outcomes will be written on the
right side of each of the branches of the tree diagram. We write the probabilities of particular
events on the branch along the path denoting the event.

Worked Out Examples

Example: 1

A coin is tossed twice. Write all the possible outcomes in a tree diagram.

Solution:
Let 'H' and 'T' be the events of getting a head and tail respectively.

1st toss 2nd toss Outcomes

1 H HH

2

H 1
T 2
1 1 T HT
2 2

H 1 H TH
T 2

1
2

1

2 T TT

Oasis School Mathematics-10 355

Example: 2

A fair coin is tossed thrice. Show all the possible outcomes is a tree diagram. Also find the
probability of getting.

(i) all three tails (ii) tail, head, tail in order
(iii) tail, tail, head in order (iv) at least two heads (v) at most two tails

Solution:

Let H and T be the events of the head and tail occurring on each toss. On tossing thrice, the
outcomes are shown in a tree diagram.

1st toss 2nd toss 3rd toss Outcome Probability

1/2 H HHH 1/2 × 1/2 × 1/2 = 1
1/2 8
H
1/2 T HHT 1/2 × 1/2 × 1/2 = 1
8

1/2 H 1/2 H HTH 1/2 × 1/2 × 1/2 = 1
1/2 T HTT 8
H THH
T T 1/2 1/2 T THT 1/2 × 1/2 × 1/2 = 1
1/2 TTH 8
1/2 1/2 H TTT
H 1/2 × 1/2 × 1/2 = 1
8
1/2 T
1/2 × 1/2 × 1/2 = 1
T 1/2 H 8

1/2 T 1/2 × 1/2 × 1/2 = 1
Fig.Tree Diagram 8

1/2 × 1/2 × 1/2 = 1
8

Now, finding the probability of

(i) P(all three tails)

i.e., P(TTT) = P(T) × P(T) × P(T)

= 1 × 1 × 1
2 2 2

= 1
8

(ii) P(tail, head, tail in order)

i.e., P(THT) = P(T) × P(H) × P(T)

= 1 × 1 × 1 = 1
(iii) P(tail, tail, head in order) 2 2 2 8

i.e., P(TTH) = P(T) × P(T) × P(H)

= 1 × 1 × 1 = 1
2 2 2 8

356 Oasis School Mathematics-10

(iv) P (at least two heads)

i.e. P(HHH or HHT or HTH or THH)

= P(HHH) + P(HHT) + P(HTH) + P(THH)

= 1 + 1 + 1 + 1
= 8 8 8 8
4 1
8 = 2

(v) P(at most two tails)

i.e. P(TTH or HTH or HHT or THH or THT or HTT or HHH)

= P(TTH) + P(HTH) + P(HHT) + P(THH) + P(THT) + P(HTT) + P(HHH)

= 1 + 1 + 1 + 1 + 1 + 1 + 1
8 8 8 8 8 8 8

= 7
or, 8

P(at most two tails) = 1 – P(TTT)

= 1– 1 . 1 . 1
2 2 2

= 1– 1
8

= 7
8

Example: 3

A bag contains 6 blue balls and 8 green balls. Two balls are Outcome
drawn in succession without replacement. By drawing a tree B BB
diagram, find the probability of getting both balls blue. G BG

Solution: B 5/13
8/13
Let B and G be the event of getting blue 6/14
and green balls respectively.
6 Blue
Total balls = 6 + 8 = 14 8 Green

A probability tree diagram is shown alongside 8/14 6/13 B
G
GB

Now, the probability of getting both balls blue is P(BB) 7/13 G GG
6 5 3 5 15 Fig.Tree Diagram

= P(B) × P(B) = 14 × 13 = 7 × 13 = 21

Oasis School Mathematics-10 357

Example: 4

Three children are born in a family. Calculate the probability of having two sons by drawing
a tree diagram.
Solution:

Let S and D denote a son and a daughter respectively. Then, probability and outcomes are
shown in a probability tree diagram as outcomes.

1st birth 2nd birth 3rd birth Outcome Probability

1/2 S SSS 1/2 × 1/2 × 1/2 = 1
1/2 8
S 1/2
1/2 D D SSD 1/2 × 1/2 × 1/2 = 1
S S 1/2 8
D 1/2
1/2 1/2 S SDS 1/2 × 1/2 × 1/2 = 1
1/2 8
1/2 D 1/2 1/2
1/2 SDS 1/2 × 1/2 × 1/2 = 1
S 8
D D
1
1/2 DSS 1/2 × 1/2 × 1/2 = 8

S 1
8
DSD 1/2 × 1/2 × 1/2 =

D

S DDS 1/2 × 1/2 × 1/2 = 1
8

D DDD 1/2 × 1/2 × 1/2 = 1
8

Fig.Tree Diagram

The probability of exactly two sons is P(SSD or SDS or DSS).

P(SSD) + P(SDS) + P(DSS) = 1 + 1 + 1 = 3
8 8 8 8

Exercise 18.3

1. A coin is tossed twice in succession. Calculate the probability that both are tails by
drawing a tree diagram.

2. A bag contains 5 red and 7 white balls. Two balls are drawn at random one after the other
without replacement. Find the probability of getting

(i) both red balls (ii) both white balls
(iii) one red ball and one white ball in any order

3. (a) Two cards are drawn in succession from a well shuffled pack of 52 cards without
replacement. Find the probability of getting

(i) both cards having the same color (ii) both cards having different colors
(b)
Two cards are drawn one after another from a pack of 52 cards without replacement.
Using a diagram, find the probability of both getting a king.

358 Oasis School Mathematics-10

4. (a) A coin is tossed thrice. Show all possible outcomes in a tree diagram with each of the
probabilities at the side of the branches. Also find the probability of getting,

(i) all three tails (ii) tails, heads, tails in order
(iii) tails, tails, heads in order (iv) at least two heads

(b) A dice is rolled twice. Using a tree diagram, find the probability of getting a '6' in both
case.

5. A family has three children. Draw a tree diagram to show all possible combinations of
boys and girls.

Calculate the probability of getting

(i) all boys (ii) all of the same sex (iii) exactly two girls (iv) exactly one boy

6. Two spanners are colored as shown.

Black Black Grey

White Grey White

'A' 'B'

(a) If both spanners are spun, draw and label a tree diagram showing all the possible

outcomes.

(b) Using a tree diagram, calculate the probability of getting,

(i) two blacks (ii) two whites (iii) a white and a grey

7. A bag contains 7 white and 5 black balls. Two balls are drawn at random, one after another
without replacement. Draw a tree diagram to show all the possible outcomes. Also find
the probability of getting,

(i) two white balls (ii) the balls of the same color (iii) balls of different colors

8. A bag contains 6 black and 4 white marbles. A marble is drawn at random from the bag
and the marble is replaced second marble is drawn. By using a tree diagram, find the
probability of getting,

(i) both of the same color and (ii) both marbles of different colors

9. (a) There are 3 sweets, one yellow, one red and one black in a bag. A sweet is taken out
randomly and not replaced. Then after, another sweet is drawn. Write the sample
space using a tree diagram. Calculate the probability of selecting one yellow and
one red sweet.

(b) There are 4 balls, 1 red, 2 blue and 1 green in a basket. Two balls are drawn one after
another without replacement. Show all the possible outcomes in a tree diagram.



Oasis School Mathematics-10 359

Answer

1 57 (iii) 35 25 26 b. (i) 1
1. 4 2. (i) 33 (ii) 22 66 3. a. (i) 52 (ii) 51 221

4. (a) (b)
H
T H 1/2 H HHH 1/6 1/6 S SS
1/2 1/2 T HHT S SS
5. 1/2 H HTH 1S S
B H 1/2 T HTT 5S 5/6
G 1/2 1/2 T 1/2 H THH
1/2 T THT 5/6
7. 1/2 1/2 H TTH
1/2 T TTT
9 (a). T 1/2 H 1 1 1/6 S SS
8 8 S S SS
(i) (ii) 1
5/6 36
1/2 (iii) 1
T 8

(iv) 1
2

6.

B 1/2 B 1/2 W WW
1/2 1/2 G 1/4 G WG
B
B 1/2 G W 1/4 B WB
1/2 1/2 G 1/2 B
1/2 G
1/2 B 1 1 1/4 1/2 W GW
1/2 G 8 4 1/4 G GG
1/2 W 1/4
1/2 B (i) (ii) G G 1/4 B
1/2 B
GB
G 1/2

1/2 (iii) 3 (iv) 3 W BW
G 8 8
1/4 G BG (b) (i) 1 ,

B 1/2 8
1/4 (iii) 3
B BB (ii) 1, 16
8

8.

W WW W WW

6/11 4/10

7/12 W B WB 4/10 W
5/11 6/10
7W 4W
5B 6B B WB

5/12 W BW (i) 7 6/10
22
7/11 4/10 W BW
B B
(ii) 31 (i) 13
4/11 66 6/10 25

B BB (iii) 35 B BB (ii) 12
66 25

Y RY (b)
1/2
B
2/3

R 1/2 B RB R 1/3 G

1/3 R YR 1/4 1/3 R
1/2 1/3 B
1Y 1/3
1R Y 1/2 B YB 2/4 B 1/3 G
1B 1/2
Y BY 1R R
1/3 1/2 2B 1/3
1G

B 1/2 BR 1 G
R 2/3 B
3

360 Oasis School Mathematics-10

Miscellaneous Exercise

1. A card is drawn from a pack of 52 cards. Find the probability of getting a red face card
[Ans: 123]
or a queen.

2. A card is drawn from a set of cards numbered 20 to 50. Find the probability of getting a

card which is the multiple of 3 or 7.
[Ans : 3113]


3. The probability of getting a scholarship by A is 1 and by B is 1 . Find the probability of
getting the scholarship by 3 4

i) both of them

ii) at least one of them. [Ans: 112, 1 , 1 ]
iii) none of them 2 2

4. A card is drawn from a pack of 52 cards. Show all the possible outcomes of getting or

not getting face cards in a tree diagram. [Ans: consult your teacher]

5. A bag contains 2 red, 2 blue and 1 green balls. Two balls are drawn one after another
without replacement. Find all the possible outcomes in a tree diagram.

[Ans: consult your teacher]

Full Marks: 12

Attempt all the questions: [6 × 2=12]

1. (a) A card is drawn from a pack of 52 cards. Find the probability of getting a red queen
or a black card.
(b)
From a set of numbered cards 5 to 20, a card is drawn. Find the probability of
(c) getting a card which is the multiple of 3 or a multiple of 4.

The probability of hitting the target by A is 1 and that of B is 1 . If both of them
try to hit the target, find the probability of 4 3

(i) hitting the target by both of them.

(ii) hitting the target by only one of them.

(iii) hitting the target by none of them.

2. (a) A bag contains 2 red, 1 blue and 1 green balls. Two balls are drawn one after another
without replacement. Show all the possible outcomes in a tree diagram and hence
find the probability of getting

(i) both balls of the same color. (ii) one red ball and another green ball.

(b) A couple gave birth to 3 children. Show all the possible outcomes in a tree diagram.

(c) Two cards are drawn from a pack of 52 cards one after another without replacement.
Show all the possible outcomes in a probability tree diagram of getting face card
and non-face card. Also find the probability of both getting a face card.

Oasis School Mathematics-10 361

Specification Grid

Subject: Compulsory Mathematics Full marks: 100 Time : 3 hours

S.N. Area/Topic Unit Knowl- Concept Applica- Higher Total Remarks
edge (K) (C) tion (A) ability (HA) Marks
1. Sets 1 Sets At least one
Tax and Money - - 1×4 1×4 4 question of
Exchange two marks
2. Arithmetic 2 Compound 1×1 should be
Interest asked from
3. Population each unit
Growth and
4. Depreciation 2×2 1×4 1×5 14

3. Mensura- 5. Plane surface 1×1 3×2 1×4 1×5 At least one
tion question of
6. Cylinder and 16 three marks
Sphere should be
7. Prism and asked from
Pyramid each unit

4. Algebra 8. HCF and LCM 1×1 5×2 2×4 1×5 At least one
5. Geometry 9. Surds and 2×1 question of
three marks
Radicands 3×2 3×4 1×5 should be
10. Indices 24 asked from
11. Algebraic each unit

fractions At least one
12. Equation question of
13. Area of 25 four marks
should be
Trianlge and asked from
parallelogram each unit
14. Construction 6
15. Circle
7
6. Trigonom- 16. Trigonometry - 1×2 1×4 - 4
37
etry 1×2 1×4 - 100
2×2 - -
7. Statistics 17. Statistics 1×1 17 10 4
34 40 20
8. Probability 18. Probability -

Total number of questions 6

Total marks 6

362 Oasis School Mathematics-10

Model Test Paper Class: x Full marks: 100

Time: 3 hours.

Attempt all the questions: 3 × (1 + 1) = 6
Group A

1 (a) If the selling price of an article and net selling price are given, write the formula
to calculate VAT percent.

(b). If the area of the base of a cylinder is A square unit and the height is 'h', find its

volume.

2. (a) Evaluate: ( 8 )-2/3
27

(b) Write the formula to calculate exact median in continuous data.

A

3. (a) In the given figure, write the relation x
between x and y. BD

A y

C

(b) In the given figure, write the relation MO
between OM and AB. B

Group B [ 4 × ( 2 + 2 ) + 3 × ( 2 + 2 + 2 = 3 4 ) ]

4. (a) A man earns $ 40.80 per hour. Using $1 = Rs. 108.40, find his daily income in
rupees if he works 8 hours a day.

(b) The present value of a machine is Rs. 45,000, what will be its value after 2 years
if the rate of depreciation be 5% p.a.

5. (a) Length of the base of an isosceles triangle is 10 cm and its area is 60 cm2, find the
length of equal sides.

(b) If three metallic spheres of radius r1 cm are melted and recast into a new sphere
having radius r2, find r1:r2.

(c) Find the total surface area of the given triangular prism. 10cm
8cm

50cm

Oasis School Mathematics-10 363

6. (a) Find the H.C.F of : a4 + a2 + 1 and a2 + a + 1

(b) Solve: x – 3 = x + 9

7. (a) Simplify: 3n+1 ÷(39n-n1+)1n-1
(3n)n-1

(b) Simplify: 2x–4 – x
x2–4 x+2

(c) The longest side of a right angled triangle is 5 cm if the length of the other two sides

are x cm and (x + 1)cm, find the length of other two sides.

AB

8. (a) In the given figure, ABCD is a square and ABFE ED F C
is a parallelogram. If AC = 10 2 cm, find the D
area of the parallelogram ABFE. A

(b) Find the value of x in the given figure. O
B 160°

x

C

(c) In the given figure, O is the centre of the circle and D O
ABC is the tangent at B, find the value of x. x cm 7cm

A 24cm B C

A B
C
9. (a) If the area of the given parallelogram ABCD is 20cm², 8cm
5cm
find the magnitude of ∠ABD

D

(b) In a grouped data the mean is 20, ∑fm = is 25a + 150, and ∑f = is 10 + a, find the
value of 'a'.

10. (a) A card is drawn from a pack of 52 cards. Find the probability of getting an ace or
a face card.

(b) A bag contains 1 red, 1 green and 1 blue ball. Two balls are drawn one after
another without replacement. Show all the possible outcomes in a tree diagram.

364 Oasis School Mathematics-10

Group C [10 × 4 = 40]
11. In a survey among 300 students, the number of students who like both subjects

mathematics and science is twice the number of students who don't like both subjects.
If the number of students who like only one subject is 120, using a Venn-diagram,
find the number of students who,
i. like mathematics
ii. like science
12. Lakpa bought a machine from China at 15000 Chinese Yuan. It is imported to Nepal
after paying 20% transportation charge and 150% customs duty. If he wants to make
the profit of 50% on his total expenditure, find the selling price of the machine in
Nepali market including 13% VAT, given that 1 Yuan = Rs. 16.25.

13. If the TSA of given square based pyramid is 384 cm2, find its
(i) slant height (ii) height (iii) volume.
14. Find the H.C.F. of: x³ – 1, x3 - 1–2x2 + 2x, x² – 3x + 2

15. The sum of the ages of a father and his son is 77 years. After 5 years the father's age
will be 3 years less than two times the age of his son, find their present ages.

16. Prove that a parallelogram and a rectangle standing on the same base and between
the same parallel lines are equal in area.

17. Verify experimentally that an inscribed angle is half of the angle at the centre standing
on the same arc.

18. Construct a ∆ABC where AB = 6.8 cm, BC=7 cm ∠ABC = 60°. Also construct a triangle
DBC equal in area with ∆ABC where ∠DBC = 90°.

19. From the top of the house 15 m high, the angle of elevation of the top of the tower is 60°.
If the distance between the house and the tower be 50 m, find the height of the tower.

20. From the given table, calculate the median.

Class interval 0-10 10-20 20-30 30-40 40-50

Frequency 4 7 12 10 6

Oasis School Mathematics-10 365

Group D [4 × 5 = 20]

21. Saleem deposited Rs. 20,000 in a bank at the rate of 10% p.a. compounded annually.
At the end of second year he withdrew Rs. 5,000 and at the end of fourth year he
deposited Rs. 15,000. Find the total interest received by him at the end of 5 years if the
interest is compounded half yearly in the last year..

22. Two pillars of a gate of a house are as shown in the figure. Find 9 ft.
the cost of painting their surface at the rate of Rs. 20 per square 2.5ft
foot.
9 ft.
2.5ft

23. If abc. = 1, prove that:

1 1 1 1.4 ft 1.4 ft
a+ 1 + b + c-1 1 + c + a-1
1 + b-1 + + =1 A
Q
24. In the given figure, ∆APQ is an isosceles triangle and ∆ABC is an C

equilateral triangle then prove that P

i. PQ||BC B

ii. ∠PAB = ∠QAC



366 Oasis School Mathematics-10