MATHEMATICS SM025
25. The time taken for 70 students to walk from the hostel to class in a certain college
are shown in the following table.
Time (minute) Number of student
2-4 5
5-7 9
8 - 10 19
11 - 13 21
14 - 16 12
17 - 19 4
(a) Find the mean and mode.
(b) Determine the 40th percentile.
(c) Find the standard deviation.
(d) Calculate the Pearson’s coefficient of skewness. Interpret your answer .
( 2015/16 )
26. The mean survival times(weeks), x , of a sample of 20 animals in a clinical
trial is 28 with summary statistics x2 18000
(a) Find the standard deviation correct to three decimal places.
(b) It is known that the median is 26, computer Pearson’s Coefficient of
Skewness. Comment on your answer. ( 2016/ 17 )
27. The following list is the number of car thefts during the year 2013 in 11
particular cities.
110 340 210 300 660 115 135 400 180 145 265
(a) Find the median.
(b) Draw a box-and whisker plot to represent the data. Hence, state the
shape of the distribution of the date and give your reason ( 2016/17 )
28. A sample of positive integers is arranged in ascending order as follows :
3x 2 , 40, 4x , 2 y , 59, 3y 9
If the mean and median of the sample are 49 and 47 respectively, determine
the values of x and y. Hence, rewrite the sample in ascending order.
( 2017/18 )
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29. The frequency distribution for 80 employees at a supermarket according to
their daily wage class is as shown below:
Class Boundary of Daily Wage(RM) Frequency
15 20 4
20 25 12
25 30 20
30 35 32
35 40 8
40 45 4
(a) Find the median and standard deviation of the sample.
(b) Calculate and interpret the Pearson’s Coefficient of skewness for the
data.
(c) Determine the daily wage k where 80% of the worker earn at most k
ringgit per day. ( 2017/18 )
Suggested Answers
1. x 84, median = 87
2. Mean = 14.72 , median = 15.25 , mode = 16.5 , standard deviation 3.46
3. (a) median = 2 (b) mean = 2.1
4. Mean = 15.8 , Q1 11.54
5. (a) 29.65 30 (b) 27.56 28
6. (a) Mean = 17 , median = 16 , mode = 12
(b) (i) Q1 12 (ii) 7.74
7. (a) w 5 (b) 45.69% , data is inconsistent
8. (a) mean = 12.2 , mode= 13.88 , median= 12.92 .
mean median mode , data is skewed to left,
(b) 24.41
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9. (a) mean = 75.23, s = 4.22
(b) 75.23 + a, variance remains the same
10. (a) i. mode = 15.91, median = 23.0
ii. mean = 26.2, standard deviation = 15.31,
skewness = 0.627 or 0.672
(b) For a skewed data, the mean is being dragged in the direction of the skew
(no more equal to the mode and median). So, it loses ability to provide the
best central tendency. Therefore, median is the best indicator of central
tendency, since it is not strongly influenced by the skewed values.
11. (a) mean = RM151.90, s = RM22.44
(b) RM155
(c) Sk = –0.138, the distribution is skewed to the left
(d) income of fishmongers are more stable because their CV is smaller
than the CV of the fruitsellers.
12. mean = 54.9, median = 55.29
13. Mean = 765 15.3 , median = 14.5, mode = 13.5
50
14. Mean= 0.9 , variance =0.8265
15. (a) median = 50.476 , mean = 51.167
(b) Sk 0.093 , the data distribution is slightly/ small positively skewed or
approximately symmetric
16. Mean = 24.8125, Mode = 23, Sk = 0.267
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17. (a) Mean = 161.88 198
(b) Q1 = 148, Q2 = 158, Q3 = 168
(c) 1118
130 148 158 168 192 *
110 120 130 140 150 160 170 180 190 200 210 220 230 240
18. x = 8, y = 9, Variance = 30.23
19. (a) mean = 10.72, median = 10.808, standard deviation = 3.812
(b) Pearson coefficient of skewness = -0.071 @ -0.072
Slightly/ small negative skewed or approximately symmetric
(c) i) mean or median because skewness is small
ii) either or both because almost symmetry
iii) median because mean approximate median
20. w = 81, mode = 51 and 59, P80 = 77.5
21. (a). mean = 3.278, mode = 1.75, median = 2.75, Positively skewed
(b). P75 = 5.1, 75% of the children are at most 5.1 years old.
22. mean=14, mode=13, standard deviation=7.266
23. (a) mode=56, median=52, Q1 36,Q3 60
(b) mean=49.44,standard dev=14.90
(c) -0.44 or -0.5154
24. Mean weight = 57.68
25. (a) 10.63 ; 11.05 (b) 9.71 (c) 3.85 (d) -0.109 (negatively
skewed)
26. (a) 11.050 (b) 0.543 , positively skewed.
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27. (a) 210
(b)
LF UF
Q1 Q2
Q3
110 130 210 340 4 660
0
Positively skewed . 0
28. x 11 , y 25 , 35, 40 , 44, 50, 59, 66
29. (a) Median = 30.6 Standard deviation = 5.85
(b) Sk 0.323 , distribution is negatively skewed.
(c) 34.4
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TOPIC 7: PERMUTATIONS AND COMBINATIONS
1. Fifteen glasses of similar colour and size, are labeled each with a number from
1 to 15. In how many ways can
(a) 8 glasses be selected such that 5 are labeled with odd numbers and the rest
with even numbers?
(b) 5 glasses be arranged in a row such that the first 3 glasses are labeled with
odd numbers and the other two with even numbers? ( 2003/04 )
2. A gymnastic team consists of 5 men and 7 women.
(a) In how many ways can the team be lined up if all the men have to be
together?
(b) Find the number of ways a team of 5 members can be formed if the team
consists of
(i) 3 men and 2 women.
(ii) At least 3 men. ( 2004/05 )
3. (a) How many words consisting of three alphabets can be formed from the
word TSUNAMI if
(i) none of the alphabets can be repeated?
(ii) every alphabets can only be used once in each word and no word starts
with M ?
(b) How many ways can two alphabets be chosen from the word TSUN and
one alphabet from the word AMI ? ( 2004/05 )
4. How many ways can all the alphabets from the word PUTRAJAYA be
arranged if
(a) the first letter is P and the last letter is A?
(b) all the letters A must be next to each other?
(c) they begin with a consonant and end with a vowel? ( 2005/06 )
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5. In a cooking competition, 6 out of 10 contestants are females. In how many
ways can one choose
(a) the champion, first and second runner-up?
(b) four winner consisting of at least two females?
(c) winners for first, second and third place consisting of two females and a
male? ( 2005/06 )
6. A total of 13 candidates comprising 5 mathematicians and 8 physicists will be
selected to form a committee. In how many ways can
(a) a committee of 5 members can be formed if it consists of at least 3
mathematicians ?
(b) All candidates be placed in a row such that they always sit in a group of
same expertise?
(c) Any 3 member may be selected from the candidates for the positions of
president, secretary and treasurer?
(d) a committee of 5 members with a mathematician as president and a
physicist as secretary can be formed? ( 2006/07 )
7. (a) There are 10 men, 15 women and 12 children participating in a family day
event.
i. In how many ways can a group of 7 men, 13 women and 10 children be
formed if a particular lady and a particular child must be in that group?
ii. Thirty participants are required in an event. In how many ways can
this group be formed if each group must consist of at least 8 men?
(b) A five-digit number may be formed from the digits 1, 2, 3, 4, 5, 6, 7 and 8
with no repetition. How many
i. five-digit numbers having values between 10000 and 50000 can be
formed?
ii. five-digit even numbers having values more than 60000 can be
formed? ( 2007/08 )
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8. (a) In how many ways can 6 women and 3 men be arranged in a row if
i. the row begins with a man and ends with a woman?
ii. the men must be separated from each other?
(b) In how many ways can a four-member committee be formed from 6
women and 3 men if the committee has equal number of both sexes?
( 2008/09 )
9. The selection committee of a competition will determine the winners for the
first to the fifth place. If twelve females and eight males participate in the
competition, in how many ways can one select
(a) three females for the first to the third place winners, and two males for the
fourth and fifth place winners?
(b) five winners which consist of three females and two males?
(c) at least four females win in the competition? ( 2008/09 )
10. (a) How many one-, two-, three-, and four-digit numbers can be formed by
using the digits 4, 5, 6, and 7, when each digit can be used only once?
(b) How many of the numbers formed in part (a) are odd and greater than
600? ( 2009/10 )
11. Three boxes A, B and C has identical green and red dice as shown in the
following table
Box A BC
Colour
Green 4 5 3
Red 5 6 4
(a) If all dice in box A are arranged in a row, in how many different
arrangements can this be done?
(b) A die is randomly drawn from each of the boxes. If all dice drawn are
of the same colour, in how many different ways can this be done?
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(c) Four dice are randomly drawn without replacement form box B. How
many different ways can the dice drawn such that there are equal
number of red and green dice ?
(d) A die randomly drawn from box A is put into box B. subsequently, a
die drawn from box B is put into box C. Finally, a die is drawn from
box C and the colour is noted. Calculate the probability that the die
drawn from C is green ( 2010/11 )
12. Six yellow balls are labeled with the numbers 1, 2, 3, 4, 5 and 6, and four red
balls are labeled with letters P, Q, R and S. All the ten balls are of similar size. In
how many different ways can one
(a) arrange all the balls in a straight line such that balls of the same colour are
next to each other?
(b)Choose and arrange equal number of yellow and red balls in a straight line
such that balls of the same colour are next to each other? ( 2011/12 )
13. A team of four members will be formed by selecting randomly from a group
consisting of four students and six lecturers. Calculate the number of different
ways to form a team consisting of
(a) No students at all
(b)Equal number of students and lecturer
(c) More students than lecturers ( 2011/12 )
14. A security code is to be formed by using three alphabets and four digits chosen
from the alphabets {a, b, c, d, e} and digits {1,2,3,4,5,6}. All the digits and
alphabets can only be used once. Find the number of different ways the security
code can be formed if
(a) there is no restriction imposed.
(b)all alphabets are next to each other and all digits are next to each other.
(c) it consists of at least two consonants. ( 2012/13 )
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15. (a) In the final of a science quiz competition, team A and B sit in rows facing
each other. Each team consists of two females and two males. Find the
number of different seating arrangements of all the contestants if
(i) in each team, contestants of the same gender request to sit next to each
other.
(ii) contestants of the same gender do not sit next to each other in team A, or
in team B, contestants of the same gender sit at both ends of the row.
(b) A test consists of ten true-false questions. How many possible arrangements
of answers can be obtained if
(i) all questions must be answered?
(ii) only six randomly chosen questions must be answered. ( 2014/15 )
16. Seven identical boxes are labeled with numbers 1,2,3,4,5,6 and 7. If five boxes
are chosen at random,
(a) find the number of different ways to arrange the boxes in a row such that
(i) there are two odd and three even numbered boxes.
(ii) there are only one even numbered box.
(b) find the probability that there are only two odd numbered boxes next to
each other. ( 2014/15 )
17. Given a set of digits 0,1,2,3,4,5,6,7,8,9.
(a) Find the number odd different ways to choose two prime digits from
the set.
(b) Four-digit numbers are to be formed from the set and the numbers do
not start with digit 0. Find the possible number of ways of getting
(i) Even numbers between 6000 and 7000 if every digit can be repeated.
(ii) Numbers greater than 6000 that end with digit 5 and the digits can only
be used only once.
(iii) Numbers that contain exactly two odd digits and they must be next to
each other with no repetitions of digits allowed. ( 2015/16 )
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18. (a) A total of 6 students can sit on 10 chairs which are arranged in a row.
(i) Find the number of different ways that all the 6 students can sit .
(ii) If both seats at the ends are o be seated, find the number of different
ways this can be done.
(iii) If 2 particular students do not sit next to each other, find the number of
different ways that all 6 students can sit.
(b) A committee consisting of 2 males and 3 females is to be formed from
5 males and 7 females. Find the number of different ways if
(i) A particular female must be in the committee.
(ii) 2 particular males cannot be in the committee. ( 2016/17 )
Suggested Answers
1. (a) 1960 (b) 14112
2. (a) 4838400 (b) (i) 210 (ii) 246
3. (a) (i) 210 (ii) 180 (b) 18
4. (a) 2520 (b) 5040 (c) 16800
5. (a) 720 (b) 185 (c) 360
6. (a) 321 (b) 9676800 (c) 1716 (d) 6600
7. (a) i) 600600
ii) 7480980
(b) i) 3360
ii) 1200
8. (a) i) 90720 ii) 151200
(b) 45
9. (a) 73920 (b) 739200 (c) 570240
10. (a) 64 (b) 18
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11. (a) 126 MATHEMATICS SM025
(c) 150
(b) 180
373
(d) or 0.4317
864
12. (a) 34560 (b) 23808
13. (a) 15 (b) 90 (c) 25
(c) 529200
14. (a) 756000 (b) 43200
(iii) 1040
15. (a) (i) 64 (ii) 16 (b) (i) 1024 (ii) 13440
16. (a) (i) 720 (ii) 360 (b) 288
17. (a) 6 (b) (i) 499 (ii) 224
18. (a) (i) 151200 (ii) 50400 (iii) 120960
(b) (i) 150 (ii) 105
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TOPIC 8: PROBABILITY
1. (a) For any sets R and S, P(R) = P(S ' R) + P(S R). Show that
PS R 1 PS' R.
(b) If P(R) 0.5 , P(S ') 0.3 and PS' R 0.4 , evaluate P(R S)
( 2003/04 )
2. A total of 120 students of a private college ride 3 types of motorcycles
(KRISS, SUZUKI and HONDA) to campus. From the total, 75 riders are
males. Out of 50 students who ride KRISS, 30 are females. There are 30
males who ride SUZUKI and 5 females who ride HONDA. If one students
who rides a motorcycle to campus is chosen at random, find the probability
that the student
(a) rides a KRISS or a SUZUKI.
(b) is female or rides a SUZUKI.
(c) rides a SUZUKI given that the student is female.
(d) is male who rides a SUZUKI or female who rides a KRISS. (2003/04 )
3. Given that the two events E and F with P(E) 0.1 and P(F) 0.3 are
independent.
(a) State a condition for two events E and F to be independent. ( 2004/05 )
(b) Find P(E F) . Are the events E and F mutually exclusive ?
(c) Find PE ' F '
4. The workers in a factory need to attend a competency course and pass three
tests. The probability of passing the first test is 0.9 and if a worker passes a
test, the probability that the worker will pass the subsequent test is 0.7.
Instead, if the worker fails, the probability that the worker will fail the
subsequent test is 0.8.
(a) Construct a tree diagram for the events.
(b) Find the probability that a worker will pass the first and the third test,
(c) Find the probability that a worker will pass at least two test. ( 2004/05 )
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5. Given A and B are two events with the following probabilities :
P(A) 2 , PA B ' 3 and PA B 1 .
77 21
(a) Find PA B.
(b) Determine whether A and B are independent events ( 2005/06 )
6. A game is involves rolling a fair die followed by drawing a marble from either
an urn A or B. If the outcome is less than 3, then a marble is drawn at random
from urn A. Otherwise, a marble is drawn at random from urn B. Urn A
contains 3 red marbles, 4 blue marbles and 3 green marbles. Urn B contains 3
red marbles and 1 blue marble.
(a) Find the probabilities that
(i) a red marble is chosen
(ii) the outcome of the die is less than 3 if it is known that the marble drawn
is red.
(iii) it is red or a green marble
(b) The similar game is repeated but two marbles are drawn instead from either
urn A or B. Find the probability that both marbles are red if the first marble
is taken
(i) with replacement
(ii) without replacement ( 2005/06 )
7. Given two events A and B with the following probabilities:
P(A) 2 , PA ' B 3 , P(A B) 1
55 15
Find
(a) P(B)
(b) PA B ' ( 2006/07 )
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8. Ali plans to buy a computer and the probability that he gets a loan is 0.6. The
probability that he will buy the computer if he gets the loan is 0.9, and the
probability that he will still buy the computer even though without getting the
loan is 0.7.
(a) What is the probability that Ali will buy a computer?
(b) If it is known that Ali did not buy a computer, what is the probability that
he failed to get the loan. ( 2006/07 )
9. The probability of a woman giving birth to a baby boy is 0.6. If the woman
gave birth to 3 children, find the probability that
(a) The number of sons exceeds the number of daughters.
(b) All three children are of the same gender. ( 2007/08 )
10. The probability that a student passes Mathematics is 0.4. If the student passes
Mathematics, the probability that the student will pass Physics is 0.7. If the
student fails Mathematics, the probability the student will pass Physics is 0.63.
(a) Calculate the probability that the student passes
i. Physics,
ii. Mathematics if the student passes Physics.
(b) Determine whether the events of a student passing Mathematics and
Physics are independent. ( 2007/08 )
11. A total of 400 new students at a college were interviewed to find out if they
either receive a scholarship, loan or no financial aid. There are 150 male
students, of which 50 receive loan and 70 do not receive any financial aid.
One hundred female students receive scholarship. There are 140 students who
do not receive any financial aid. If a new student is selected at random,
calculate the probability that the student is a
(a) Female or a scholarship recipient.
(b) Loan recipient if it is known that the student is a female.
(c) Male who is a scholarship recipient or a female who receive a loan.
(d) Female or non-scholarship recipient. ( 2008/09 )
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12. Given two events A and B with P(B) 1 , P(A B) 3 , P(B A) 1
3 44
Find
a) P(A)
b) P(A ' B ') ( 2009/10 )
13. In chess tounament between A and B, the probability A wins is 0.2, B wins is
0.5 and the probability of a draw is 0.3. If A and B were to meet in three
games, calculate the probability that
(a) Two games are draw
(b) A and B win alternately
(c) Either A or B wins all the games
(d) B wins at least two games ( 2010/11 )
14. Given P(A) 0.5 , P(B) 0.6 and P(A B) 0.8 . Calculate the probability
that both events A and B occur. hence, verify that A and B are independent
events. ( 2011/12 )
15. It is found that 30% of the population of an island are overweight. Among the
overweight, the probability of those who do not have any chronic illness is 0.4
and among those who are not overweight, the probability that they do not have
any chronic illness is 0.65.
Draw a tree diagram to represent the given information.
(a) Hence, if a person is randomly chosen from that population, find the
probability that he
(i) does not have any chronic illness.
(ii) is overweight knowing that he does not have any chronic illness.
(b) If two persons are randomly chosen from the population, find the
probability that
(i) both of them do not have any chronic illness.
(ii) only one of them has chronic illness.
(iii) at least one of them does not have any chronic illness
( 2011/12 )
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16. A box consists of five grape-flavoured sweets and four strawberry-flavoured
sweets. All the sweets are of the same size. A child chooses at random four
sweets from the box. Find the probability that
(a) all sweets are of the same flavor.
(b) less than three sweets are strawberry-flavoured. ( 2012/13 )
17. Every year two teams, Unggul and Bestari meet each other in a debate
competition. Past results show that in years when Unggul win, the probability
of them winning the next year is 0.6 and in years when Bestari win, the
probability of them winning the next year is 0.5. It is not possible for the
competition to result in a tie. Unggul won the competition in 2011.
(a) Construct a probability tree diagram for the three years up to 2014
(b) Find the probability that Bestari will win in 2014
(c) If Bestari wins in 2014, find the probability that it will be their first
win for at least three years.
(d) Assuming that Bestari wins in 2014, find the smallest value of n such
that the probability of Unggul wins the debate competition for n consecutive
years after 2014 is less than 0.05 ( 2012/13 )
18. The events A and B are independent with P(A) = x, P(B) = x + 0.3 and
P(A B) 0.04. Determine the value of x. Hence, find P(A B)'. ( 2013/14 )
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19. The probability that a person is a carrier of Thalassemia is 0.03. If a person is
actually a carrier, the probability a medical diagnostic test will give a positive
result, indicating that he is a carrier, is 0.92. If the person is actually not a
carrier, the probability of a positive result is 0.03. Draw a tree diagram to
represent the given information.
(a) A medical diagnostic test is said to be efficient if 5% of the time it
gives a correct positive result. Determine if the test is efficient
(b) Find the probability that the test gives a negative result.
(c) What is the probability that the diagnostic test gives a negative result
and the person tested is not a carrier?
d) Two persons who took the test are randomly chosen. What is the
probability that both give positive results? ( 2013/14 )
20. Given P(A B') 0.25, P(A) 0.48 and P(B) 0.42 . Find P(A B).
Is A and B mutually exclusive events? Hence, determine whether A and B are
independent events. ( 2014/15 )
21. In a college there are 150 students taking courses in Chemistry, Physics and
Biology.
Among the students, 92 are females. There are 48 students taking Chemistry
which 28 are females. Half of the 68 students taking Physics are females.
(a) Construct the contingency table for the given data.
(b) A student is chosen at random. Find the probability that the student
(i) takes Biology.
(ii) is a male, given that he takes Biology
(iii) takes Biology or a female.
(c) Two students are chosen at random, find the probability at least one
student is a female and takes Biology. ( 2014/15 )
22. Given P(A) 0.37, P(B A) 0.13 and P(A'B) 0.47 . Find ( 2015/16 )
(a) P(A B) .
(b) P(B) and hence calculate P((A B)') .
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23. A car insurance company offers two types of insurance plan for privately-
owned cars, namely Plan X and Plan Y. for a random sample of 60 clients for
each insurance plan, the number of claims is given in the following table.
Claim
Plan Yes No
X 38 22
Y 45 15
Let
A = the event that no claim is made by the client
B = the event that the customer takes Plan X.
(a) Find P(A B) .
(b) Find P(A'B) .
(c) Given that the chosen client did not make any claim, find the
probability that the insurance plan taken was Plan X.
(d) Determine whether the events ‘make a claim’ and ‘the type of each
insurance plan taken’ are independent. Give reason for your answer.
( 2015/16 )
24. Given that P(A) 0.35 and P(B) 0.45 .Calculate
(a) P(A B) if event A and B are mutually exclusive.
(b) P(A B ') if event A and B are independent. ( 2016/17 )
25. The table below shows the classification of 200shirt based on sizes and
colours.
White Small Medium Large
Blue 40 35 5
Black 10 30 15
25 20 20
A shirt is selected randomly. Find the probability that the shirt is ( 2016/17 )
(a) Small in size.
(b) Either blue or white.
(c) Medium size given that it is blue.
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26. Let A and B be two events where P(A ') 0.7 , P(B) 0.4 and P(A B) 0.6 .
Determine P(A B) and then evaluate P A B . Hence, state with reason
whether A and B are independent events. ( 2017/18 )
27. A survey was implemented on 400 students at a private university in order to
collect information on the popular choice of minor subjects ( Language,
Statistics and Information Technology ) by students of various major of study
( Medicine, Engineering and Economics ). The following tables describes the
data collected from the survey.
Major Medicine Engineering Economics Total
Minor
30 80 30 140
Language 10 30 10 50
Statistics 60 120 30 210
InformationTechnology 100 230 70 400
Total
If a student from this group is selected at random, what is the probability that
he
(a) Is either majoring in Medicine or doing a minor in Information
Technology?
(b) Is a non-Medicine student who does a minor in Language?
(c) Chooses a minor in Statistics knowing that he is an Economics student ?
(d) Is neither an Engineering student who does a minor in Statistics nor is he
an Economics student who does a minor in Language? ( 2017/18 )
Suggested Answers
1. (a) DIY (b) 0.9
2. (a) 3 (b) 5 (c) 2 (d) 1
4 8 9 2
3. (a) P(E F) P(E) P(F) or PF E P(F) or PE F P(E)
(b) P(E F) 0.03 , E and F are not mutually excluscive because P(E F) 0
(c) 0.9 .
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4. (a) 0.7 P3
0.9
0.1 0.7 P2 0.3 F3
(b) 0.495 P1 0.2 P3
0.3 F2 0.8 F3
0.2 0.7 P3
F1 P2
0.3 F3
0.8 F2 0.2 P3
(c) 0.698 0.8 F3
5. (a) 43 (a) not independent.
63
6. (a) (i) 3 (ii) 1 (iii) 7
5 6 10
(b) (i) 81 (ii) 16
200 45
7. (a) 1 (b) 2
6 5
8. (a) 0.82 (b) 2
3
9. (a) 0.648 (b) 0.28
10. (a) i) 0.658 ii) 0.426 (b) Not independent
11. (a) 0.7 (b) 0.32 ( c) 0.275 (d) 0.925
12. (a) 5 (b) 3
9 8
13. (a) 0.189 (b) 0.07 (c) 0.133 (d) 0.5
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14. 0.3
15. C
0.6
O
0.3 0.4 C
C
0.7 0.35
O
0.65 C
(ii) 0.209
(a) (i) 0.575 (ii) 0.489 (iii) 0.819
(b) (i) 0.331
16. (a) 0.048 (b) 0.833
17. (a)
(b) 0.444 (c) 0.3243 (d) 6
18. 0.54
19. (a). 0.0567. The test is efficient. (b). 0.9433 (c). 0.9409 (d). 0.00321
20. P(A B) 0.23, not mutually exclusive events
A and B are not independent events.
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21. (a) CPB T
58
M 20 34 4 92
F 150
T 28 34 30
48 68 34
(b) (i) 0.2267 (ii) 0.1176 (iii) 0.64
(c) 0.3609
22. (a) 0.0481 (b) 0.5181; 0.16
23. (a) 11 (b) 7 (c) 22 (d) not independent
60 8 37
24. (a) 0.80 (b) 0.193
25. (a) 3 (b) 27 (c) 6
8 40 11
26. P(A B) 0.1 , PA B 0.25
27. (a) 5 (b) 11 (c) 1 (d) 17
8 40 7 20
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TOPIC 9: RANDOM VARIABLES
1. The probability distribution function of a random variable X is
P( X x) k 3 for x = 0, 1, 2, 3
x
where k is a constant. Determine
(a) the value of k. Hence, find P(0 X 3) .
(b) E(X ) , Var(X ) and Var(2X 3) . ( 2003/04 )
2. The probability density function of a continuous random variable X is given by
f (x) cx 2 0 x 3
0 others
where c is a constant. Find
(a) the value of c and P(1 X 2)
(b) the median of X.
(c) E(X ) and Var(X ) . ( 2003/04 )
3. The probability density function of a continuous random variable X is given by
f (x) x 0 x 1
2x 1 x k
otherwise
0
where k is a positive constant.
(a) Show that k 2
(b) Calculate P(0.5 X 3)
(c) Find the mean and variance of X. ( 2004/05 )
4. A discrete random variable X has a probability distribution function given by
P(X x) m 3 x , x 0,1,2,3,4,5
2
where m is a constant.
(a) Determine the value of m.
(b) Calculate P(X x), x 0,1,2,3,4,5 .
(c) Find the mean and variance for Y 5 2X . ( 2004/05 )
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8. The probability function of a discrete random variable X is
P(X = x) = kx
2
where x = 1, 2, 3, 4, 5, 6 and k is a constant.
(a) Show that k 2 .
21
(b) Find VarX .
9. The time taken by a student (in hours) to study is given by a continuous variable
X, with a cumulative density function of
0 if x0
F (x) = 1 k(10 x)2 if 0 x 10
1 if x 10
where k is a constant.
a) Determine the value of k
b) Find P(3 X 9)
c) Determine the probability density function of X for 0 x 10.
d) Find the median of X.
e) Obtain the variance of X ( 2007/08 )
10. A bookstore recorded the number of books, X, sold daily with the probability
P(X x) 6 x , X 0,1, 2,3, 4 .
20
Calculate E(X ) . Hence, find P X E(X ) 3 . ( 2008/09 )
2
11. A continuous random variable X has the cumulative function given by
0 , x 1
x a2 , 1 x 3
F(x) 12
where a and b are constants.
14x x2 25 , 3 x 7
b
1 , x7
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(a) Show that a 1 and b 24
(b) Find P(2 X 5) .
(c) Calculate the median.
(d) Determine the density function.
(e) Sketch the graph of f (x) and hence find the mode.
12. The following table represents the probability distribution of a discrete random
variable Y.
y −2 −1 1 3 5
P(Y = y) 0.1 0.3 0.4 0.1 0.1
Find
a) P(Y 1).
b) E(Y 3)2 and VarY 3. ( 2009/10 )
13. The probability density function of a continuous random variable X is given by
k ln x , 1 x e
x
f ( x)
0, otherwise
Show that k = 2.
Hence,
a) Obtain the cumulative distribution function, F(x).
b) Determine the 81st percentile for the distribution of X.
c) Calculate E(X). ( 2009/10 )
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14. The cumulative probability distribution function of a discrete random variable X
is
0, x 1
1 x 2
1 ,
2 x4
6 4 x6
2 x6
P( X x) ,
3
5 ,
6
1,
(a) Construct the probability distribution table of X.
(b) Calculate Var (X). ( 2010/11 )
15. A continuous random variable, Y has a probability density function
c(3 y 2 ), 0 y2
f ( y) 0, otherwise
Show that c 3 and E(Y ) 3 .
10 5
Hence,
a) Calculate P(Y E(Y )).
b) Determine the mode of the distribution.
c) Show that the median, m of the distribution satisfies the equation
m3 9m 5 0. ( 2010/11 )
16. A nurse works five days in a week. The number of days in a week she works
overtime is a discrete random variable X with probability function.
k 3x 1, x 0,1,2
3 x 3,4
k (x 2), x5
f (x)
3
k (x 1),
3
where k is a constant. Show that k 1
5
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(a) Find the probability she works overtime everyday in a week
(b) Calculate the probability that she will work overtime for at least three days in a
week.
(c) Determine the most likely number of days in week she will work overtime.
(d) Find the expected number of day in week she will work overtime. Hence,
evaluate E(3X 1) . ( 2011/12 )
17. The continuous random variable X has the cumulative distribution function
0 , x0 where a is constant.
F (x) a x3 3x2 , 0 x 1
, x 1
1
Show that a 4. Hence,
(a) Calculate the mean and variance of X.
(b) Find PX E( X ) 110.
(c) If Y 4X 3, find the E(Y ) and Var(Y ). ( 2011/12 )
18. A discrete random variable X has a probability distribution function
25x , x 1, 2, 3, 4
p(x) 32 where k is a constant.
k , x5
(a) Show that k 1 .
16
(b) Find P1 X 3.
(c) Calculate the mean of X and hence, calculate E 2X 3.
(d) Find the variance of X and hence, calculate Var 9 2X . ( 2012/13 )
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19. A continuous random variable X has the probability density function
6x , 0 x 1
5
6
f ( x) 2 x 2 , 1 x2
5
0 , otherwise
(a) Find the cumulative distribution function of X.
(b) Find
(i) P0.5 X 1.5.
(ii) P X 1.5.
(c) Calculate the median of X correct to three decimal places. ( 2012/13 )
20. The probability distribution function of a discrete random variable X is given
as follows:
X 123456
P(X x) 1 1 3 2 5 3
kkkkkk
where k is a constant. Determine the value of k. Hence, calculate Var(2X 1) .
( 2013/14 )
21. The number of vehicles owned by residents in a housing estate, X is a discrete
random variable with probability distribution function
f (x) 1x21x01, , x0
x 1, 2, 3, 4.
(a) Verify that f(x) is a probability distribution function.
(b) Find the probability that a resident has more than two vehicles.
(c) Find the cumulative distribution function for X and hence determine the
median.
(d) Let Y = 30X + 10 be the monthly fee (in RM) imposed by a security
company. Find the expected amount of monthly fee to be paid.
( 2013/14 )
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22. The time for patients to experience complications in a week after a heart
surgery, X is a continuous random variable with probability density function
given by
f (x) a(1 x), 0 x 1
0, otherwise.
where a is a constant. Show that a = 2.
Hence, find the
(a) cumulative distribution function of X and estimate its median.
(b) mean and variance of X. Calculate Var(3 2X ) . ( 2013/14 )
23. Let X be the random variable representing the number obtained when a biased
dice is rolled. The probability of the biased dice to give odd numbers is three
times higher than even numbers when it is rolled.
(a) If the dice is rolled once,
(i) construct a probability distribution table for X.
(ii) find the probability of getting a number less than 3.
(iii) find the mean and variance of X.
(b) If the dice is rolled 100 times, find the expected value of getting the
number 6. ( 2014/15 )
24. The cumulative distribution function of a continuous random variable, X is
given as follows:
0, x0
F (x) 3121,x(x 4), 0 x4
x4
(a) Calculate P( X 1 1).
(b) Find the median. ( 2014/15 )
(c) Determine the probability density function of X.
Hence, evaluate E(3X 2 1).
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25. The continuous random variable X has cumulative distribution function F(x)
given by
0, x0
0 x2
F ( x) x2 x2 ,
6 2 x3
2x 2, x3
3
1,
Find
(a) P(1 X 2.2) .
(b) The value of median.
(c) The probability density function of X.
(d) The expected value of X.
(e) The variance of X, given that E( X 2 ) 19 . ( 2015/16 )
6
26. Two dice are thrown and the numbers x and y obtained from each dice are
noted. The discrete random variable W is defined as
W xy, x y
x y, x y
(a) Write all the outcomes for W 4 and hence show that
P(W 4) 5 .
36
(b) Construct a table of the probability distribution of the random variable
W. hence, show that W is a discrete random variable.
(c) Find P(W 9) .
(d) Find the mode of W.
(e) Find E(W) and hence, calculate E(3 - 4W). ( 2015/16 )
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27. The number of times, X, a certain statistics book is borrowed from a library
per semester is modeled as probability distribution function below
P(X x) k(7 2x) x 0,1,2,3
0 otherwise
with k as a constant. Find k. Hence,
(a) Construct a probability distribution table for X.
(b) Find P(X 2)
(c) Calculate E(2X 3) .
(d) Find Var(2X 3) ( 2016/17 )
28. Let probability density function of a continuous random variable X be defined
by
f (x) x2 , c x c
18
0, elsewhere
With c is a constant.
(a) Show that c 3.
(b) Find the cumulative distribution function of X.
(c) Hence, find
(i) P(0 X 2)
(ii) The median of X. ( 2016/17 )
29. A game is conducted by tossing a biased coin 3 times. The coin has
probability P(H ) 1 and P(T ) 2 , where the event in obtaining head is H
33
and the event in obtaining tail is T.
(a) Construct a tree diagram and hence, show that the probability of
getting one head is 12 .
27
(b) Let X be the number of heads that appears, find the probability
distribution of X.
(c) Suppose a player wins RM 2 each time a tail appears.
(i) Find the probability distribution of Y.
(ii) Calculate E(Y ) and Var(Y ) . ( 2016/17 )
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30. A probability distribution for discrete random variable X is as shown in the
table below
X 2 1 0 1 2 3
P(X x) p 0.1 0.3 q 0.2 p q
Where p and q are constants. If E(X ) 0.65, determine the values of p and q.
Hence, calculate the standard deviation of X. ( 2017/18 )
31. The probability distribution function of a discrete random variable X is given
as
P(X x) x , x 1,2,3
17
x , x 4,5,6,7
34
0, otherwise
(a) Calculate P(2 X 5) .
(b) Determine the value of Var(X ) . Hence, calculate the standard
deviation ofY 5X 1 ( 2017/18 )
32. Continuous random variable X has a density probability function is given by
f (x) ax, 0 x 1
1 x 4
a (4 x), otherwise
3
0,
Where a is a constant.
(a) Find the value of a.
(b) Find the E(X ) and Var(2 3X ) .
(c) Evaluate PX E(X ) a
(d) Estimate the median. ( 2017/18 )
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SUGGESTED ANSWERS
1. (a) k 1 , 3 (b) E(X ) 3 , Var(X ) 3 , Var(2X 3) 3
84 24
2. (a) c 1 , P(1 X 2) 7 (b) 2.38
9 27
(c) E(X ) 9 , Var(X ) 27
4 80
3. (a) DIY (b) 0.875 (c) E(X ) 1 , Var (X ) 1
6
4. (a) m 1
10
(b)
x 012 34 5
35
P(X x) 3 1 1 20 20 7
20
20 20 20 (iii) 14.5
(b) E(X ) 67 , Var(5 2X ) 12.51
20
0, x 1
1, 1 x 0
8
5. (a) F (x) 4, 0 x 1
8
6, 1 x 2
8
7, 2 x3
8
1, x3
(b) (i) E(X ) 3 (ii) median =0
4
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6. (a) 2 (b) median 0.630 , E( X ) 3
4 5
7. (a) k 1
5
(b) 0, x 1
F(x) 1 x 2
x3 1,
5 2 x4
x 1, x4
5
1,
(c) (i) Var(X ) 1.251 (ii) Var(3X 2) 11.25 (iii) 13
15
8. ( a) DIY (b) 20
9. a) k = 1 9
100 b) 0.48 c) f (x) 1 (10 x), 0 x 10
50
d) m = 2.929 0,
otherwise
e) Var(X ) 50
9
10. 3 ; 9
2 20
11. (b) 3 (c) 3.54
4
0.33
(d) f (x) x 1, , 1 x 3
6 (e)
7 3 x7
12 x elsewhere
12
0,
13 7
Mode = 3
12. (a) 0.3 (b) 9.3 ; 4.01
0, x 1
13. a) F (x) ln x2 , 1 x e b) e0.9 c) 2
1,
xe
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14. (a) 4 6
X -1 2
1
P(X x) 1 3 1 6
666
(b) 4.5833
15. (a) 0.482 (b) mode = 0
16. (a) 4 (b) 7 (c) 2 (d) E(X ) 43 , E(3X 1) 48
15 15 15 5
17. (a) E(X ) 2 ; Var(X ) 19 (b) 0.664
5 350
(c) E(Y ) 1.4 ; Var(Y ) 152
175
18. (a) k 1 3 31 7
16 (b) (c) ;
367 367 4 16 8
(d) ;
256 64
19.
, x0
0 , 0 x 1
, 1 x 2
, x2
3 x2
(a) F (x) 5
22 x3
1
5
1
(b) (i) 4 (ii) 1 (c) 0.913
5 20
20. k 15 , Var(2X 1) 8.64 Page 87
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MATHEMATICS SM025
0 , x0
1
, 0 x 1
120 , 1 x 2
61 , 2x3
21. (b).0.1583 (c). F(x) 112001 , median = 1 (d). 241 60.25
4
120
29
, 3 x 4
30 , x4
1
0 , x0
22. (a). F(x) = 2x x2 , 0 x 1 , median = 0.293
1 , x 1
(b). E( X ) 1 , Var(X ) 0.056 , Var(3 2X ) 0.222
3
23. (a)(i)
X 12345 6
P(X x) 3 1 3 1 3 1
12
12 12 12 12 12
(ii) 1 (iii) mean 13 , var iance 137
3 4 48
(b) 8.33
24. (a) 0.375 (b) m 2.47
(c) f (x) x2 , 0 x4 , E(3X 2 1) 19
16
0, otherwise
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x , 0 x2
3
25. (a) 31 2x 2 x3
50 (b) 3 (c) f (x) 3 2, otherwise
(e) 7
(d) 5 0,
3 18
26. (a) 2,2,5,1,1,5,2,6,6,2; P(W 4) 5
36
(b)
W 1 2 3 4 5 9 16 25 36
P(W w) 11 8 6 5 2 1 1 1 1
36 36 36 36 36 36 36 36 36
P(W w) 1
(c) 1 (d) 1 (e) 161 ; 134
12 36 9
27. k 1 0 1 23
16
7 5 31
(a) 16 16 16 16
x
P(X x)
(b) 15 (c) 19 (d) 55
16 4 16
28. (a) DIY (b) F (x) 0, x 3
(c) (i) 4 3 x3
27 x3 1 ,
54 2 x3
1,
(ii) median 0
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29. (a)
(b)
x0 1 23
P(X x) 8 12 6 1
27 27 27 27
(c) (i) 0 2 46
y
1 6 12 8
P(Y y) 27 27 27 27
(ii) E(Y ) 504 , Var(Y ) 2.667
27
30 p 0.15 , q 0.05 , standard deviation = 1.71
31 (a) 7 (b) Var(X ) 1041 , standard deviation = 4.24
17 289
32. (a) a 1 (b) E( X ) 5 , Var(2 3X ) 13
2 32
(c) 0.7199 (d) median 1.55
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TOPIC 10 : SPECIAL PROBABILITY DISTRIBUTION
1. The average number of students entering the main door of a library between
9:00 to 9:30 am during the school holidays is 10.
(a) Calculate the probability that 3 to 5 students will enter the library at that
particular time.
(b) If the probability of less than m students entering the library at that time is
0.583, find the value of m. ( 2003/04 )
2. The monthly earnings of operators in a particular factory are normally
distributed with mean RM780 and standard deviation RM8.
(a) If the factory has 900 operators, how many operators earn between
RM770 to RM800 a month?
(b) If 67% of the operators earn more than RMd monthly, what is the
value of d? ( 2003/04 )
3. The probability that a person is cured from pneumonia after being given a new
type of medicine is 0.4.
(a) If a sample of 20 patients is randomly selected,
(i) find the mean and standard deviation of patients that will be cured.
(ii) find the probability that 4 to 12 patients will be cured.
(b) If a sample of 100 patients is randomly selected, find the probability that
less than 66 patients will not be cured. ( 2003/04 )
4. The distribution of the number of car breakdowns on a highway in any one
day is Poisson with mean 3.5.
(a) Find the probability that
(i) exactly 2 cars break down on a particular day.
(ii) At most 5 cars break down on a particular day.
(iii) between 100 to 111 cars break down on the days for the month of
April.
(b) An auto repair company places 3 of its trucks to provide assistance in car
breakdowns along the above highway everyday. Find the probability that on a
particular day the company could not provide any such assistance ( 2004/05 )
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5. It is know that 10 % of the patients with high fever are confirmed to be
suffering from dengue fever.
(a) If 15 patients with high fever are random chosen, find the probability that
(i) less than 9 are confirmed to be suffering from the dengue fever.
(ii) exactly 10 patients with high fever are confirmed to be free of dengue
fever
( b) If 100 patients with high fever are randomly chosen,
(i) approximate the probability that 9 to 14 patients are confirmed to be
suffering from dengue fever.
(ii) find the value of m such that the probability of more than m patients
that are confirmed to be suffering from dengue fever is 0.025.
( 2004/05 )
6. The number of short messages(SMS) received by a teenager in half an hour
has a Poisson distribution with the mean .
(a) If the probability of received no SMS within half an hour is 0.0025, show
that 6 (to the nearest integer)
(b) using the value of 6 , find the probability that
(i) he receives less than six SMS in half an hour
(ii) he receives less than six SMS in one hour.
(iii) Two teenagers selected at random will receive at least six SMS in half
an hour ( 2005/06 )
7. In any of its shipments, a company found that the probability of bad oranges it
supplies is 0.2. At the receiving terminal, a sample is taken at random and the
number of bad oranges is recorded.
(a) A shipment will be rejected if there are more than 10% bad oranges in the
sample taken. Calculate the probability that a particular shipment will be
accepted if a sample of size 20 is taken.
(b) Using the normal approximation, estimate the probability of obtaining 180
to 210 bad oranges if 1000 oranges are inspected at random.
(c) In another shipment, the probability of obtaining bad oranges is 0.03. The
probability of rejecting this shipment is 0.022. Using the Poisson
approximation, determine the maximum allowable number of bad oranges
in a sample of size 300 such that the shipment is accepted. ( 2005/06 )
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8. In a delivery of microchips, it is known that number of defective is 2 out of 10.
(a) If 15 microchips are delivered, calculate the probability that
(i) at least 5 microchips are defective
(ii) Exactly 11 microchips are good .
(b) If 500 microchips are delivered, find n such that the probability of
obtaining the number of defective microchips exceeding n is 0.147
(c) Suppose in another shipment of microchips, the probability of defective is
0.01. If a sample of 300 microchips is taken from the shipment, estimate
the probability of getting 1 to 3 defective microchips. ( 2006/07 )
9. Assume that the number of e-mails received by a student daily has a Poisson
distribution with mean of 5.
a) i. Determine the probability that the student receives between 5 and 13
e-mails daily.
ii. If the probability of a student receiving not more than m e-mails in a
day is 0.616, determine the value of m.
b) If 15 days are randomly chosen, find the probability that the students
receives between 5 and 13 e-mails daily for a period of 9 days.
c) If 150 days are randomly chosen, use the normal approximation to find the
probability that the student receives between 5 and 13 e-mails daily for
less than 70 days. ( 2007/08 )
10. Compact discs produced by a factory are packed in boxes. Each box contains
100 compact discs. It is known that 4% of the compact discs produced are
defective.
(a) Show that the probability that a box chosen at random will contain at most
3 defective compact discs is approximately 0.43.
(b) Find the probability that among 12 boxes chosen at random, there will be 4
boxes which contain at most 3 defective compact discs.
(c) Seventy boxes are chosen at random. Find the probability that between 20
boxes and 40 boxes, inclusively, which contain at most 3 defective compact
discs. ( 2008/09 )
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11. In any large shipment of watermelons from a particular orchard, it is known
that 2% are unripe. Upon arrival shipment at a receiving depot, random
sampling with replacement are conducted.
a) Calculate the probability of getting at most one unripe watermelon in a
sample of size 20.
b) If the sample size is 1000, approximate the probability of getting not more
than eight unripe watermelons. ( 2009/10 )
12. The distribution of the weights of all sugar sachets produced by a particular
factory is assumed to be normal with mean 25 gm and the standard deviation
2 gm.
Show that the probability of a randomly selected sachet weighs within 1gm of
the mean is 0.383.
a) If ten sachets are randomly selected, find the probability that between four
and seven weigh within 1gm of the mean.
b) Determine the sample size, n such that the probability that none of the
sachet weighs within 1gm of the mean is 0.021.
c) If one hundred sachets are randomly selected, approximate the probability
that less than 40 sachets weigh within 1gm of the mean. ( 2010/11 )
13. On an average, a hospital receives 6 emergency calls in 15 minutes. It is
assumed that the number of emergency calls received follows the Poisson
distribution.
(a) Find the probability that not more than 15 emergency calls are received
in an hour.
(b) Find the number of emergency calls received, m, if it is known that the
probability at most m emergency calls received in half an hour is
0.155. ( 2011/12 )
14. The probability that a type of antibiotics can cure a certain disease is 0.95.
(a) If five patients are given the antibiotics, find the probability that
(i) exactly three patients are cured after finishing the course of
antibiotics.
(ii) at least one patient is cured after finishing the course of antibiotics.
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(b) If 500 patients are given the antibiotics, find the
(i) probability that more than 480 patients are cured.
(ii) largest possible value n such that the probability that at least n
patients recovered after finishing the course of antibiotics is 0.9.
( 2011/12 )
15. The number of motorcycles arriving at the main entrance of a university
during peak hours has a Poisson distribution with mean three per minute. Find
the probability that
(a) at most one motorcycle will arrive in one minute.
(b) exactly five motorcycles will arrive in two minutes. ( 2012/13 )
16. The registration record of a private college indicates that 40% of its new
intakes are international students and the remaining are local students.
(a) If 20 new students are randomly selected and the number of local
students are noted, find the probability that there are
(i) equal number of local and international students.
(ii) not less than 9 local students.
(b) Exactly 100 new students are randomly selected. By using a suitable
approximate distribution,
(i) find the probability that between 38 and 46 are international
students.
(ii) determine the value m such that the probability that the number
of international students is at most m is 0.993. ( 2012/13 )
17. A discrete random variable X has a Poisson distribution with parameter . By
using its probability distribution function, show that
PX y 1 P(X y) where y is an integer. Given that E(X) = 1.5,
y 1
find P(X = 2). ( 2013/14 )
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MATHEMATICS SM025
18. The lifetime of D sized batteries produced by a local factory is normally
distributed with mean 11.5 months and standard deviation 0.8 months.
(a) Suppose a battery is selected at random from the factory’s production
line.
(i) Calculate the probability that the battery’s lifetime is between
9.5 and 11.5 months, correct to one decimal place.
(ii) If the probability that the battery’s lifetime is less than h
months is 0.975, determine the value of h.
(b) Suppose ten batteries are selected at random from the factory’s
production line, calculate the probability that at most three batteries
have lifetime between 9.5 and 11.5 months.
(c) If 100 batteries are selected at random from the factory’s production
line, approximate the probability that from 48 to 51 batteries have
lifetime between 9.5 and 11.5 months. ( 2013/14 )
19. A survey found that 32% of teenage consumers earned their spending money
from working part-time. If five teenagers are selected at random, find the
probability that at least two of them are working part-time. ( 2014/15 )
20. Number of accidents at a particular location of a highway occurs at the rate of
1.6 per week. Find the probability
(a) there will be two accidents in a week.
(b) there are more than 10 accidents in a five weeks period. ( 2014/15 )
21. An egg is classified as grade A if it weighs at least 100 grams. Suppose eggs
lay at a particular farm has the probability of 0.4 being classified as grade A
eggs.
(a) If 15 eggs are selected at random from the farm, calculate the
probability that more than 20% of them are not grade A eggs.
(b) A retailer bought 500 eggs from the farm.
(i) Approximate the percentage that the retailer would have bought
from 220 to 230 grade A eggs.
(ii) If the probability not more than m of the eggs bought are of
grade A is 0.9956, determine the value m. ( 2014/15 )
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MATHEMATICS SM025
22. The length of newborn babies at a hospital for a particular year is normally
distributed with mean of 52 cm and standard deviation of 2.5 cm. a baby’s
length is considered normal if it is between 46 cm and 56 cm. from a list of
100 birth records selected randomly for that particular year at the hospital,
how many babies are expected to have normal lengths? ( 2015/16 )
23. A car rental company has 7 cars available for rental each day. Assuming that
each rental is for the whole day and that the number of demands has a mean of
3 cars per day. Find the probability that
(a) The company cannot meet the demand in any one day.
(b) Less than 5 cars are rented in a period of 3 days. ( 2015/16 )
24. It is known that 37% of the students at a college do not take breakfast
regularly. A random sample of 20 students is chosen.
(a) Find the probability that there are at least two students who do not take
breakfast regularly.
(b) Use normal approximation to calculate the probability that there are
more than 10 students who do not take breakfast regularly. Verify that
the distribution can be approximated by a normal distribution.
( 2015/16 )
25. For every class of 40 students on average there are 4 of them are left-handed.
Find the probability that
(a) Exactly 5 students are left –handed in any class.
(b) Between 4 and 17 students are left-handed in any two classes. ( 2016/17 )
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MATHEMATICS SM025
26. The amount of grains packed in a sack is normally distributed with mean
weight and standard deviation 6 kg. Given P(X 24) 0.1587 . The sack
is separated from others if it weight less than 25 kg.
(a) Find the value of .
(b) Hence,
(i) Find the probability that a randomly chosen sack has weights of more
than 33 kg.
(ii) Find the probability that a randomly chosen will be separated.
(c) A total of 5 sacks are chosen at random, find the probability that
(i) All the sacks are to be separated.
(ii) At least 4 of the sacks are to be separated. ( 2016/17 )
27. The number of text messages received by Rosnaida during a fixed time
interval is distributed with a mean of 6 messages per hour.
(a) Find the probability that Rosnaida will receive exactly 8 text messages
between 16:00 and 18:00 on a particular day.
(b) It is known that Rosnaida has received at least 10 text messages between
16:00 and 18:00 on a particular day, find the probability that she received
13 text messages during that time interval. ( 2017/18 )
28. In each delivery of cupcakes to a particular restaurant, 30% will be returned
due to not favoured by cupcakes lovers.
(a) Suppose 20 of the cupcakes are randomly selected from a delivery, what is
the probability that at most 5 will be returned ?
(b) Suppose the restaurant will be holding an event which requires an order of
200 cupcakes from the same supplier.
(i) Approximate the probability that between 56 and 62 of the cupcakes
will be returned.
(ii) If the probability of observing less than n number of cupcakes among
those delivered which are returned is 0.992, use the normal
approximation to determine the value of n. ( 2017/18 )
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MATHEMATICS SM025
Suggested Answers
1. (a) 0.0643 (b) m 11
2. (a) 799 (b) d RM 776.48
3. (a) (i) 8 , 2.19 (ii) 0.963 (b) 0.8686
4. (a) (i) 0.1850 (ii) 0.8576 (iii) 0.3754 (b) 0.4634
5. (a) (i) 0.9978 (ii) 0.0105 (b) ( i) 0.6247 (ii) m 15
6. (a) 6 (b) (i) 0.4457 (ii) 0.0203 (iii) 0.3072
7. (a) 0.2061 (b) 0.7441 (c) k 15
8. (a) (i) 0.1642 (ii) 0.1876 ( b) 109 (c) 0.5982
9. (a) (i) 0.382 (ii) m 5 (b) 0.0483 (c) 0.9798
10. (a) DIY (b) 0.1886 (c) 0.9887
11. (a) 0.9401 (b) 0.00466
12. (a) 0.2818 (b) n 8 (c) 0.5987
13. (a) 0.0344 (b) m 8
14. (a) (i) 0.0214 (ii) 1
(b) (i) 0.1292 (ii) n 469
15. (a) 0.199 (b) 0.1606
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MATHEMATICS SM025
16. (a) (i) 0.1172 (ii) 0.9435
(b) (i) 0.4903 (ii) m = 52
17. 0.251
18. (a) (i) 0.5 (ii) h = 13.068 (b) 0.1719 (c). 0.3094
19. 0.5125
20. (a) 0.2584 or 0.2585 (b) 0.1841
21. (a) 0.9981 (b) (i) 3.5% (ii) m 228
22. 94
23. (a) 0.0119 (b) 0.055
24. (a) 0.9988 (b) 0.0749
25. (a) 0.1563 (b) 0.867
26. (a) 30 (b) (i) 0.3085 (ii) 0.2033
(c) (i) 0.0003473 (ii) 0.00715
27. (a) 0.0655 (b) 0.139
28. (a) 0.4164 (b) (i) 0.2964 (ii) n 76
BKS2018/19 Page 100
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