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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
Multiple-choice questions
1 A fair die is to be rolled 90 times. If X is the number of times a six is observed then the
expected value of X, E(X), is equal to:
A 1
B 3.5
C 6
D 12.5
E 15
2 Suppose that X is the number of female children born into a family. If the distribution of X
is binomial, with probability of success of 0.48, then the probability that a family with five
children will have two female children is:
A 0.48  2
B (0.48)2
C (0.48)2(0.52)3
D 5C2(0.48)2(0.52)3
E 5C2(0.52)2(0.48)3
3 Each day a random sample of 50 CDs is selected from a production process and inspected.
If on average one out of every hundred CDs is faulty, then the standard deviation of the
number of faulty CDs is closest to:
A 0.010
B 0.495
C 0.500
D 0.704
E 0.990
4 A 10-sided die has four black sides and six red sides, and each side has an equal chance of
showing. If the die is tossed four times, the probability of the same colour showing at least
three times is closest to:
A 0.1792
B 0.2800
C 0.4752
D 0.4992
E 0.6544
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
5 Which one of the following random variables has a binomial distribution?
A The number of tails observed when a coin is tossed three times
B The number of times a die is rolled before a six is observed
C The weight in kilograms of a randomly chosen student
D The time a person waits to be served at a bank
E The number of accidents that occur at an intersection in a 1-week period
6 A sample of 20 marbles is selected with replacement from an urn containing 4 black
marbles and 16 red marbles. If X is the number of black balls in the sample, then
approximately 99.0% of the time:
A X8
B X9
C X  10
D X  11
E X  12
7 Bronwyn has determined that her team’s chance of goaling from a corner in hockey is 0.3.
The probability that the team will score at least one goal from 10 corners is:
A (0.3)10
B
10
C1(0.3)1(0.7)9
C 1 – 10C0(0.3)0(0.7)10
D 1 – 10C0(0.3)0(0.7)10 – 10C1(0.3)1(0.7)9
E (0.3)10 + 10C1(0.3)1(0.7)9
8 If the proportion of batteries that are defective is 0.003, then the standard deviation of the
number of defective batteries in a sample of size 1000 is closest to (assume a binomial
distribution):
A 0.3
B 1.706
C 1.729
D 1.732
E 2.991
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
9 For a particular binomial distribution with n independent trials, each with probability of
success p, the mean and variance are 5 and 4 1 respectively. Which of the following gives
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the correct values for n and p?
A n = 20 and p = 1
6
B n = 30 and p = 5
6
C n = 20 and p = 5
6
D n = 20 and p = 1
5
E n = 30 and p = 1
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The following information relates to Questions 10, 11 and 12.
Suppose the random variable X represents the number of heads recorded when a fair coin is
tossed 100 times.
10 The mean and the variance of X are equal to
A E(X) = 50, Var(X) = 25
B E(X) = 50, Var(X) = 5
C E(X) = 100, Var(X) = 50
D E(X) = 50, Var(X) = 5
E E(X) = 25, Var(X) = 25
11 The probability exactly 50 heads are observed in 100 tosses is closest to
A 0.0796
B 0.1123
C 0.4602
D 0.5398
E 0.6178
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
12 The probability 50 or more heads are observed in 100 tosses is closest to
A 0.0796
B 0.1123
C 0.4602
D 0.5398
E 0.6178
13 Which one of the following graphs best represents the shape of a binomial probability
distribution of the random variable X with eight independent trials and probability of
success 0.3?
A
B
C
D
E
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
14 The probability of winning a prize in game of chance is 0.2. The least number of games
that must be played to ensure that the probability of winning at least one prize is more than
0.90 is closest to:
A 3
B 11
C 14
D 22
E 25
15 The probability of winning a prize in game of chance is 0.2. The fewest number of games
that must be played to ensure that the probability of winning at least two prizes is more
than 0.95 is closest to:
A 5
B 10
C 20
D 22
E 25
Short-answer questions (technology-free)
1 If X is a binomial random variable with parameters n = 3 and p =
1
, find:
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a Pr(X = 0)
b Pr(X = 1)
c Pr(X  1)
d Pr(X  1)
2 Three people are to be randomly chosen from the crowd at a football finals match to
receive prizes in a competition. If 70% of the crowd are St Kilda supporters, find:
a the probability that at least one of the winners is a St Kilda supporter
b the mean and the variance of the number of St Kilda supporters amongst the prizewinners
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
3 The audience at a special preview screening of a movie was asked to rate the movie as
‘good’, ‘OK’, or ‘awful’. If 10 of the 100 people at the screening rate the movie as ‘awful’
find the probability at least one of these people will be included in a sample of three
people chosen to comment on the movie.
4 If a panel of two people is selected at random from the population, 60% of whom are aged
25 years or less, find:
a the probability that the panel contains the same number of people aged 25 years or less
and people aged over 25 years
b the mean and variance of the number of people aged 25 years or less on the panel
5 James rolls a fair die n times.
a Write down an expression for the smallest number of rolls for which the probability of
obtaining at least one six is greater than 0.95.
b Write down an expression for the smallest number of rolls for which the probability of
obtaining at least two sixes is greater than 0.95.
6 The probability of contracting a certain disease is known to be 0.2. If there are 2500
students at a university, find the interval [ – 2,  – 2], and interpret in this context.
7 A binomial random variable is such that the mean is 10 and the variance is 7.5. Find the
value of the probability of success p.
8 An experiment consists of six independent trials, each of which can result in a success or a
failure. The probability of success is a trial is p. Find in terms of p the probability of
exactly one success, given at least one success.
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
Extended-response questions
In order to assess the quality of computer games being produced, the manufacturer selects 10
games at random before they are put into boxes for delivery to shops, and inspects them.
1 If it is known that 8% of games are defective, find, correct to four decimal places:
a the probability that one of the games in the sample will be defective
b the probability that at least one of the games in the sample will be defective
c the probability that exactly one of the games in the sample will be defective, given that
at least one of the games is defective
d the expected number of defective games in the sample
e the standard deviation of the number of defective games in the sample
2 It is found after a new machine is installed that the probability that exactly one of the
games in a sample of size 10 being defective, given that at least one of the games is
defective is equal to 0.56. What proportion of defective games is now being produced?
(Give your answer correct to two decimal places.)
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
Answers to Chapter 15 Test A
Answers to multiple-choice questions
1 E
2 D
3 D
4 E
5 A
6 A
7 C
8 C
9 E
10 A
11 A
12 D
13 C
14 B
15 D
Answers to short-answer questions (technology-free)
1 a
64
125
b
48
125
c
112
125
d
61
125
2 a 0.973
b E(X) = 2.1, Var(X) = 0.63
3 0.271
4 a 0.48
b E(X) = 1.2, Var(X) = 0.48
5
6
n
5
6
n 1
5 a    0.05
b  
 n5

  0.05
 6 
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Essential Mathematical Methods 3 & 4 CAS
Chapter 15 The binomial distribution
6 (460, 540). There is a probability of about 0.95 that from 460 to 540 students will contract
the disease.
7 0.25
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6 p(1  p) 5
1  (1  p) 6
Answers to extended-response questions
1 a 0.3777
b 0.5656
c 0.6678
d 0.8
e 0.8579
2 0.11
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