Chapter 3

In order to get at least once a head with probability \(\geq 0.9\), the number of times a coin needs to be tossed is (a) 3 (b) 4 (c) 5 (d) None of these

One bag contains 5 white and 4 black balls. Another bag contains 7 white and 9 black balls. A ball is transferred from the first bag to the second bag and then a ball is drawn from the second bag. The probability that the ball is white is (a) \(\frac{8}{17}\) (b) \(\frac{40}{153}\) (c) \(\frac{5}{9}\) (d) \(\frac{4}{9}\)

The probability of India winning a test match against West Indies is \(1 / 2 .\) Assuming independence from match to match, the probability that in a 5 match series India's second win occurs at the third test is (a) \(2 / 3\) (b) \(1 / 2\) (c) \(1 / 4\) (d) \(1 / 8\)

Three groups \(A, B, C\) are competing for positions on the Board of Directors of a company. The probabilities of their winning are \(0.5\), \(0.3,0.2\), respectively. If the group \(A\) wins, the probability of introducing a new product is \(0.7\) and the corresponding probabilities for group \(B\) and \(C\) are \(0.6\) and \(0.5\), respectively. The probability that the new product will be introduced is (a) \(0.18\) (b) \(0.35\) (c) \(0.10\) (d) \(0.63\)

A die is thrown three times. Getting a 3 or a 6 is considered a success. Then the probability of at least 2 successes is (a) \(2 / 9\) (b) \(7 / 27\) (c) \(1 / 27\) (d) None of these

An unbiased die with faces marked \(1,2,3\), 4,5 and 6 is rolled four times. Out of four face values obtained the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is (a) \(16 / 81\) (b) \(1 / 81\) (c) \(80 / 81\) (d) \(65 / 81\)

A man and his wife appear for an interview for two posts. The probability of the husband's selection is \(1 / 7\) and that of the wife's selection is \(1 / 5\). What is the probability that only one of them be selected? (a) \(1 / 7\) (b) \(2 / 7\) (c) \(3 / 7\) (d) None of these

India plays two matches each with West Indies and Australia. In any match the probabilities of India getting point 0,1 and 2 are \(0.45,0.05\) and \(0.50\), respectively. Assuming that the outcomes are independents, the probability of India getting at least 7 points is (a) \(0.8750\) (b) \(0.0875\) (c) \(0.0625\) (d) \(0.0250\)

A purse contains 4 copper coins and 3 silver coins, the second purse contains 6 copper coins and 2 silver coins. If a coin is drawn out of any purse, then the probability that it is a copper coin is (a) \(4 / 7\) (b) \(3 / 4\) (c) \(37 / 56\) (d) None of these

If out of 20 consecutive whole numbers 2 are chosen at random, then the probability that their sum is odd is (a) \(5 / 19\) (b) \(10 / 19\) (c) \(9 / 19\) (d) None of these

A man takes a step forwards with probability \(0.4\) and backwards with probability \(0.6\). Find the probability that at the end of 11 steps he is one step away from the starting point. (a) \({ }^{11} C_{6}(0.4)^{6}(0.6)^{5}\) (c) \(462(0.24)^{5}\) (b) \({ }^{11} C_{6}(0.6)^{6}(0.5)^{5}\) (d) None of these

In a box of 10 electric bulbs, 2 are defective. Two bulbs are selected at random one after the other from the box. The first bulb after selection being put back in the box before making the second selection. The probability that both the bulbs are without defect is (a) \(9 / 25\) (c) \(4 / 5\) (b) \(16 / 25\) (d) \(8 / 25\)

A cricket team has 15 members; out of which only 5 can bowl. If the names of the 15 members are put into a hat and 11 drawn random, then the chance of obtaining an 11 containing at least 3 bowlers is (a) \(7 / 13\) (b) \(11 / 15\) (c) \(12 / 13\) (d) None of these

A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or is a nail? (a) \(3 / 16\) (c) \(11 / 16\) (b) \(5 / 16\) (d) \(14 / 16\)

If two events \(A\) and \(B\) are such that \(P(\bar{A})=0.3, P(B)=0.4\) and \(P(A \bar{B})=0.5\) then \(P\left(\frac{B}{A \cup \bar{B}}\right)\) equals (a) \(1 / 2\) (b) \(1 / 4\) (c) \(1 / 3\) (d) None of these

For two events \(A\) and \(B\), if \(P(A)=P\left(\frac{A}{B}\right)=\frac{1}{4}\) and \(P\left(\frac{B}{A}\right)=\frac{1}{2}\), then (a) \(A\) and \(B\) are independent (b) \(P\left(\frac{A^{\prime}}{B}\right)=\frac{3}{4}\) (c) \(P\left(\frac{B^{\prime}}{A^{\prime}}\right)=\frac{1}{2}\) (d) All of these

Two dice are thrown together four times. The probability that both dice will show the same number twice is (a) \(1 / 3\) (b) \(25 / 36\) (c) \(25 / 216\) (d) None

If from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, 1 ball is drawn at random from each box, then the probability that 2 white and 1 black ball will be drawn is (a) \(13 / 32\) (b) \(1 / 4\) (c) \(1 / 32\) (d) \(3 / 16\)

Two dice are thrown thrice. The probability of getting at most twice equal numbers on dice is (a) \(1 / 6\) (b) \(5 / 72\) (c) \(215 / 216\) (d) None

A bag contains 4 white, 5 red and 6 black balls. If 2 balls are drawn at random, then the probability that one of them is white is (a) \(44 / 105\) (b) \(11 / 105\) (c) \(11 / 21\) (d) None of these

One card is drawn from the pack of cards. It is now replace in the pack and a card is again drawn. If it is done six times, then the probability of coming 2 cards of heart, 2 of diamond and 2 red cards in order is (a) \((1 / 4)^{4}\) (b) \((1 / 4)^{5}\) (c) \((1 / 4)^{6}\) (d) None of these

If a coin be tossed \(n\) times, then probability that the head comes odd number of times is (a) \(1 / 2\) (c) \(1 / 2^{n-1}\) (d) None of these

Six positive and 8 negative numbers are given. If 4 numbers are chosen and multiplied, then the probability of getting a positive product is (a) \(15 / 1001\) (b) \(70 / 1001\) (c) \(420 / 1001\) (d) \(505 / 1001\)

Two events \(A\) and \(B\) are such that \(P(A)=1 / 4\) \(P(B / A)=1 / 2, P(A / B)=1 / 4\), then \(P(\bar{A} / \bar{B})\) is equal to (a) \(1 / 4\) (b) \(3 / 4\) (c) \(1 / 2\) (d) \(2 / 3\)

A die is thrown four times. The probability of getting at most two 6 is (a) \(0.984\) (b) \(0.802\) (c) \(0.621\) (d) \(0.721\)

Team \(A\) has probability \(2 / 3\) of winning whenever it plays. Suppose \(A\) plays 4 games; then the probability that \(A\) wins more than half of its games is (a) \(16 / 27\) (b) \(19 / 27\) (c) \(19 / 81\) (d) \(32 / 81\)

Given that \(P(A)=1 / 3, P(B)=1 / 4, P(A \mid B)=\) \(1 / 6\), then \(P(B \mid A)\) equal to (a) \(1 / 4\) (b) \(1 / 8\) (c) \(3 / 4\) (d) \(1 / 2\)

The mean and variance of a binomial variates \(X\) are 2 and 1 , respectively. The probability that \(X\) takes a values greater than 1 is (a) \(1 / 16\) (b) \(5 / 16\) (c) \(11 / 16\) (d) \(15 / 16\)

If bag \(A\) contains 2 white and 3 red balls and bag \(B\) contains 4 white and 5 red balls. A ball is selected randomly from a randomly selected bag and is found to be red. Then the probability that it is selected from bag \(B\) is (a) \(25 / 52\) (c) \(21 / 52\) (b) \(5 / 18\) (d) \(13 / 18\)

In binomial probability distribution, mean is 3 and standard deviation is \(3 / 2\). Then the probability distribution is (a) \(\left(\frac{3}{4}+\frac{1}{4}\right)^{12}\) (b) \(\left(\frac{1}{4}+\frac{3}{4}\right)^{12}\) (c) \(\left(\frac{1}{4}+\frac{3}{4}\right)^{9}\) (d) \(\left(\frac{3}{4}+\frac{1}{4}\right)^{9}\)

The probability that \(A\) speaks truth is \(4 / 5\) and the probability that \(B\) speaks truth is \(3 / 4\). The probability that they contradict each other when asked to speak on a fact is (a) \(3 / 10\) (b) \(7 / 20\) (c) \(1 / 4\) (d) \(2 / 5\)

A box contains 3 white and 2 red balls. If the first drawn ball is not replaced, then the probability that the second drawn ball will be red is (a) \(8 / 25\) (b) \(2 / 5\) (c) \(3 / 5\) (d) \(21 / 25\)

The mean and the variance of a binomial distribution are 4 and 2, respectively, then the probability of 2 successes is (a) \(28 / 256\) (b) \(42 / 256\) (c) \(56 / 256\) (d) \(72 / 256\)

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is (a) \(1 / 3\) (b) \(1 / 6\) (c) \(1 / 2\) (d) \(1 / 4\)

If three students \(A, B, C\) can solve a problem with probabilities \(1 / 3,1 / 4\) and \(1 / 5\) respectively, then the probability that the problem will be solved is a) \(3 / 5\) (b) \(4 / 5\) (c) \(2 / 5\) (d) \(47 / 60\)

There are \(n\) urns each containing \((n+1)\) balls such that the \(i^{\text {th }}\) urn contains \(i\) white balls and \((n+1-i)\) red balls. Let \(u_{i}\) be the event of selecting the \(i^{\text {th }}\) urn, \(i=1,2,3 \ldots, n\), and \(W\) denote the event of getting a white ball. (i) If \(P\left(u_{j}\right) \alpha i\), where \(i=1,2,3, \ldots, n\), then \(\lim _{n \rightarrow \infty} P(W)\) is equal to (a) 1 (b) \(1 / 4\) (c) \(2 / 3\) (d) \(3 / 4\)