Page No 538:
Question 5:
If P(A)
, P(B) =
and P(A ∪ B) =
, find
(i) P(A ∩ B) (ii) P(A|B) (iii) P(B|A)
Page No 538:
Question 6:
A coin is tossed three times, where
(i) E: head on third toss, F: heads on first two tosses
(ii) E: at least two heads, F: at most two heads
(iii) E: at most two tails, F: at least one tail
Page No 539:
Question 7:
Two coins are tossed once, where
(i) E: tail appears on one coin, F: one coin shows head
(ii) E: not tail appears, F: no head appears
Page No 539:
Question 8:
A die is thrown three times,
E: 4 appears on the third toss, F: 6 and 5 appears respectively on first two tosses
Page No 539:
Question 9:
Mother, father and son line up at random for a family picture
E: son on one end, F: father in middle
Page No 539:
Question 12:
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?
Page No 539:
Question 13:
An instructor has a question bank consisting of 300 easy True/False questions, 200 difficult True/False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?
Page No 540:
Question 17:
If A and B are events such that P (A|B) = P(B|A), then
(A) A ⊂ B but A ≠ B (B) A = B
(C) A ∩ B = Φ (D) P(A) = P(B)
Page No 546:
Question 5:
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?
Page No 547:
Question 10:
Events A and B are such that
. State whether A and B are independent?
Page No 547:
Question 15:
One card is drawn at random from a well shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
(i) E: ‘the card drawn is a spade’
F: ‘the card drawn is an ace’
(ii) E: ‘the card drawn is black’
F: ‘the card drawn is a king’
(iii) E: ‘the card drawn is a king or queen’
F: ‘the card drawn is a queen or jack’
Page No 548:
Question 17:
The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(A) 0 (B) 
(C) 
(D) 
Page No 548:
Question 18:
Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) 
(C) P(A) = P(B)
(D) P(A) + P(B) = 1
Page No 555:
Question 1:
An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?
Page No 556:
Question 3:
Of the students in a college, it is known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an A grade, what is the probability that the student is hostler?
Page No 556:
Question 4:
In answering a question on a multiple choice test, a student either knows the answer or guesses. Let 
be the probability that he knows the answer and 
be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability What is the probability that the student knows the answer given that he answered it correctly?
Page No 556:
Question 5:
A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (that is, if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
Page No 556:
Question 6:
There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
Page No 556:
Question 7:
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
Page No 556:
Question 8:
A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that was produced by machine B?
Page No 557:
Question 10:
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3 or 4 with the die?
Page No 557:
Question 11:
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that was produced by A?
Page No 557:
Question 14:
If A and B are two events such that A ⊂ B and P (B) ≠ 0, then which of the following is correct?
A. 
B. 
C. 
D. None of these
Page No 570:
Question 2:
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represents the number of black balls. What are the possible values of X? Is X a random variable?
Page No 570:
Question 3:
Let X represents the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Page No 570:
Question 4:
Find the probability distribution of
(i) number of heads in two tosses of a coin
(ii) number of tails in the simultaneous tosses of three coins
(iii) number of heads in four tosses of a coin
Page No 570:
Question 5:
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
Page No 571:
Question 9:
The random variable X has probability distribution P(X) of the following form, where k is some number:
(a) Determine the value of k.
(b) Find P(X < 2), P(X ≥ 2), P(X ≥ 2).
Page No 571:
Question 14:
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? Find mean, variance and standard deviation of X.
Page No 571:
Question 16:
The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is
(A) 1 (B) 2 (C) 5 (D) 
Page No 576:
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?
Page No 577:
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?
Page No 577:
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?
Page No 577:
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?
Page No 577:
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)
Page No 577:
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?
Page No 577:
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?
Page No 578:
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?
Page No 582:
Question 3:
Suppose that 5% of men and 0.25% of women have grey hair. A haired person is selected at random. What is the probability of this person being male?
Assume that there are equal number of males and females.
Page No 582:
Question 4:
Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?
Page No 583:
Question 6:
In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear each hurdle is
. What is the probability that he will knock down fewer than 2 hurdles?
Page No 583:
Question 8:
If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?
Page No 583:
Question 10:
How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?
Page No 583:
Question 12:
Suppose we have four boxes. A, B, C and D containing coloured marbles as given below:
Box | Marble colour | ||
Red | White | Black | |
A | 1 | 6 | 3 |
B | 6 | 2 | 2 |
C | 8 | 1 | 1 |
D | 0 | 6 | 4 |
Page No 584:
Question 13:
Assume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
Page No 584:
Question 14:
If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability
).
Page No 584:
Question 15:
An electronic assembly consists of two subsystems, say, A and B. From previous testing procedures, the following probabilities are assumed to be known:
P(A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities
(i) P(A fails| B has failed) (ii) P(A fails alone)
Page No 584:
Question 18:
If P (A|B) > P (A), then which of the following is correct:
(A) P (B|A) < P (B) (B) P (A ∩ B) < P (A).P (B)
(C) P (B|A) > P (B) (D) P (B|A) = P (B)
Page No 584:
Question 19:
If A and B are any two events such that P (A) + P (B) − P (A and B) = P (A), then
(A) P (B|A) = 1 (B) P (A|B) = 1
(C) P (B|A) = 0 (D) P (A|B) = 0
and P(A|B) =
. Are E and F independent?
