EXERCISE 7.5
Question 1:
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes? (ii) at least 5 successes?
(iii) at most 5 successes?
ANSWER:
The repeated tosses of a die are Bernoulli trials. Let X denote the number of successes of getting odd numbers in an experiment of 6 trials.
Probability of getting an odd number in a single throw of a die is, 
X has a binomial distribution.
Therefore, P (X = x) = 
(i) P (5 successes) = P (X = 5)
(ii) P(at least 5 successes) = P(X ≥ 5)
(iii) P (at most 5 successes) = P(X ≤ 5)
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Question 2:
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.
ANSWER:
The repeated tosses of a pair of dice are Bernoulli trials. Let X denote the number of times of getting doublets in an experiment of throwing two dice simultaneously four times.
Probability of getting doublets in a single throw of the pair of dice is
Clearly, X has the binomial distribution with n = 4, 
∴ P (2 successes) = P (X = 2)
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Question 3:
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?
ANSWER:
Let X denote the number of defective items in a sample of 10 items drawn successively. Since the drawing is done with replacement, the trials are Bernoulli trials.
X has a binomial distribution with n = 10 and
P(X = x) =
P (not more than 1 defective item) = P (X ≤ 1)
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Question 4:
Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is a spade?
ANSWER:
Let X represent the number of spade cards among the five cards drawn. Since the drawing of card is with replacement, the trials are Bernoulli trials.
In a well shuffled deck of 52 cards, there are 13 spade cards.
X has a binomial distribution with n = 5 and
(i) P (all five cards are spades) = P(X = 5)
(ii) P (only 3 cards are spades) = P(X = 3)
(iii) P (none is a spade) = P(X = 0)
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Question 5:
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. What is the probability that out of 5 such bulbs
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one
will fuse after 150 days of use.
ANSWER:
Let X represent the number of bulbs that will fuse after 150 days of use in an experiment of 5 trials. The trials are Bernoulli trials.
It is given that, p = 0.05
X has a binomial distribution with n = 5 and p = 0.05
(i) P (none) = P(X = 0)
(ii) P (not more than one) = P(X ≤ 1)
(iii) P (more than 1) = P(X > 1)
(iv) P (at least one) = P(X ≥ 1)
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Question 6:
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
ANSWER:
Let X denote the number of balls marked with the digit 0 among the 4 balls drawn.
Since the balls are drawn with replacement, the trials are Bernoulli trials.
X has a binomial distribution with n = 4 and
P (none marked with 0) = P (X = 0)
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Question 7:
In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’; if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.
ANSWER:
Let X represent the number of correctly answered questions out of 20 questions.
The repeated tosses of a coin are Bernoulli trails. Since “head” on a coin represents the true answer and “tail” represents the false answer, the correctly answered questions are Bernoulli trials.
∴ p = 
X has a binomial distribution with n = 20 and p =
P (at least 12 questions answered correctly) = P(X ≥ 12)
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Question 8:
Suppose X has a binomial distribution
. Show that X = 3 is the most likely outcome.
(Hint: P(X = 3) is the maximum among all P (xi), xi = 0, 1, 2, 3, 4, 5, 6)
ANSWER:
X is the random variable whose binomial distribution is
.
Therefore, n = 6 and
It can be seen that P(X = x) will be maximum, if 
will be maximum.
The value of
is maximum. Therefore, for x = 3, P(X = x) is maximum.
Thus, X = 3 is the most likely outcome.
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Question 9:
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
ANSWER:
The repeated guessing of correct answers from multiple choice questions are Bernoulli trials. Let X represent the number of correct answers by guessing in the set of 5 multiple choice questions.
Probability of getting a correct answer is, p
Clearly, X has a binomial distribution with n = 5 and p
P (guessing more than 4 correct answers) = P(X ≥ 4)
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Question 10:
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is
. What is the probability that he will in a prize (a) at least once (b) exactly once (c) at least twice?
ANSWER:
Let X represent the number of winning prizes in 50 lotteries. The trials are Bernoulli trials.
Clearly, X has a binomial distribution with n = 50 and
(a) P (winning at least once) = P (X ≥ 1)
(b) P (winning exactly once) = P(X = 1)
(c) P (at least twice) = P(X ≥ 2)
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Question 11:
Find the probability of getting 5 exactly twice in 7 throws of a die.
ANSWER:
The repeated tossing of a die are Bernoulli trials. Let X represent the number of times of getting 5 in 7 throws of the die.
Probability of getting 5 in a single throw of the die, p 
Clearly, X has the probability distribution with n = 7 and p
P (getting 5 exactly twice) = P(X = 2)
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Question 12:
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
ANSWER:
The repeated tossing of the die are Bernoulli trials. Let X represent the number of times of getting sixes in 6 throws of the die.
Probability of getting six in a single throw of die, p
Clearly, X has a binomial distribution with n = 6
P (at most 2 sixes) = P(X ≤ 2)
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Question 13:
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?
ANSWER:
The repeated selections of articles in a random sample space are Bernoulli trails. Let X denote the number of times of selecting defective articles in a random sample space of 12 articles.
Clearly, X has a binomial distribution with n = 12 and p = 10% =
P (selecting 9 defective articles) =
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Question 14:
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(A) 10−1
(B) 
(C) 
(D) 
ANSWER:
The repeated selections of defective bulbs from a box are Bernoulli trials. Let X denote the number of defective bulbs out of a sample of 5 bulbs.
Probability of getting a defective bulb, p
Clearly, X has a binomial distribution with n = 5 and 
P (none of the bulbs is defective) = P(X = 0)
The correct answer is C.
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Question 15:
The probability that a student is not a swimmer is
. Then the probability that out of five students, four are swimmers is
(A) 
(B) 
(C) 
(D) None of these
ANSWER:
The repeated selection of students who are swimmers are Bernoulli trials. Let X denote the number of students, out of 5 students, who are swimmers.
Probability of students who are not swimmers, q
Clearly, X has a binomial distribution with n = 5 and 
P (four students are swimmers) = P(X = 4) 
Therefore, the correct answer is A.
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