Chapter 24

Assertion: \(A\) set \(X\) contains \(n\) elements. Two subsets \(A\) and \(B\) of \(X\) are chosen at random. The probability that \(A\) and \(B\) have same number of elements is \(\frac{1 \cdot 3 \cdot 5 \ldots(2 n-1)}{2^{n}(n !)}\)\begin{aligned} &\text { Reason: }\left({ }^{n} C_{0}\right)^{2}+\left({ }^{(-} C_{1}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2} \\ &=\frac{2^{n}[1 \cdot 3 \cdot 5 \ldots(2 n-1)]}{\underline{\phantom{xx}}} \end{aligned}

Assertion: A bag contains \(n+1\) coins. It is known that one of these coins has a head on both sides while the other coins are fair. One coin is selected at random and tossed. If head turns up, the probability that the selected coin was fair, is \(\frac{n}{n+2}\) Reason: If an event \(A\) occurs with two mutually exclusive and exhanstive events \(E_{1}\) and \(E_{2}\), then \(P(E / A)\) \(=\frac{P\left(E_{i}\right) P\left(A / E_{i}\right)}{P\left(E_{1}\right) P\left(A / E_{1}\right)+P\left(E_{2}\right) P\left(A / E_{2}\right)}, i=1,2 .\)

Assertion: A player tosses a coin and scores 1 point for a head and 2 points for a tail. He plays on until his score reaches or passes \(n .\) If \(P\) denotes the probability of getting a score of exactly \(n\), then \(P_{\mathrm{n}}+\frac{1}{2} P_{\mathrm{n}-1}=P_{2}+\frac{1}{2} P_{1}\) Reason: \(P_{\mathrm{a}}=\frac{1}{2}\left(P_{\mathrm{n}-1}+P_{\mathrm{n}-2}\right), n \geq 3\)

\(A\) and \(B\) play a game where each is asked to select a number from 1 to 25 . If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial, is: \(\quad\) [2002] (A) \(\frac{1}{25}\) (B) \(\frac{24}{25}\) (C) \(\frac{2}{25}\) (D) none of these

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 match series India's second win occurs at the third test is : [2002] (A) \(\frac{1}{8}\) (B) \(\frac{1}{4}\) (C) \(\frac{1}{2}\) (D) \(\frac{1}{3}\)

Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is [2003] (A) \(\frac{4}{5}\) (B) \(\frac{3}{5}\) (C) \(\frac{1}{5}\) (D) \(\frac{2}{5}\)

The mean and variance of a random variable having a binomial distribution are 4 and 2 respectively, then \(P(X=1)\) is (A) \(\frac{1}{32}\) (B) \(\frac{1}{16}\) (C) \(\frac{1}{8}\) (D) \(\frac{1}{4}\)

The probability that \(A\) speaks truth is \(\frac{4}{5}\), while this probability for \(B\) is \(\frac{3}{4}\). The probability that they contradict each other when asked to speak on a fact is \([\mathbf{2 0 0 4}]\) (A) \(\frac{3}{20}\) (B) \(\frac{1}{5}\) (C) \(\frac{7}{20}\) (D) \(\frac{4}{5}\)

The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is (A) \(\frac{37}{256}\) (B) \(\frac{219}{256}\) (C) \(\frac{128}{256}\) (D) \(\frac{28}{256}\)

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is [2005] (A) \(\frac{2}{9}\) (B) \(\frac{1}{9}\) (C) \(\frac{8}{9}\) (D) \(\frac{7}{9}\)

Let \(A\) and \(B\) be two events such that \(P(\overline{A \cup B})=\frac{1}{6}, P(A \cap B)=\frac{1}{4}\) and \(P(\bar{A})=\frac{1}{4}\), where \(\bar{A}\) stands for complement of event \(A\). Then events \(A\) and \(B\) are (A) equally likely and mutually exclusive (B) equally likely but not independent (C) independent but not equally likely (D) mutually exclusive and independent

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 -minute time intervals. The probability that there is at the most one phone call during a 10 -minute time period is [2006] (A) \(\frac{6}{5^{c}}\) (B) \(\frac{5}{6}\) (C) \(\frac{6}{55}\) (D) \(\frac{6}{e^{5}}\)

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is [2007] (A) \(1 / 729\) (B) \(\underline{8} / 9\) (C) \(8 / 729\) (D) \(8 / 243\)

Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are \(0.3\) and \(0.2\), respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is \([2007]\)(A) \(0.06\) (B) \(0.14\) (C) \(0.2\) (D) \(0.7\)

A die is thrown. Let A be the event that the number obtained is greater than \(3 .\) Let \(\mathrm{B}\) be the event that the number obtained is less than \(5 .\) Then \(P(A \cup B)\) is [2008] (A) \(\frac{3}{5}\) (B) 0 (C) 1 (D) \(\frac{2}{5}\)

One ticket is selected at random from 50 tickets numbered \(00,01,02, \ldots .49\). Then the probability that the sum of the digits on the selected ticket is 8 , given that the product of these digits is zero, equals [2009] (A) \(\frac{1}{14}\) (B) \(\frac{1}{7}\) (C) \(\frac{5}{14}\) (D) \(\frac{1}{50}\)

An urn contains nine balls of which three are red, four are blue and two are green. The experiment is to draw three balls at random without replacement from the urn. The probability that the three balls have different color is (A) \(\frac{2}{7}\) (B) \(\frac{1}{21}\) (C) \(\frac{2}{23}\) (D) \(\frac{1}{3}\)

Consider 5 independent Bernoulli's trials each with probability of failure \(1-p .\) If the probability of at least one failure is greater than or equal to \(\frac{31}{32}\), then \(p\) lies in the interval (A) \(\left(\frac{3}{4}, \frac{11}{12}\right)\) (B) \(\left[0, \frac{1}{2}\right]\)(C) \(\left[\frac{11}{12}, 1\right]\) (D) \(\left[\frac{1}{2}, \frac{3}{4}\right]\)

If \(C\) and \(D\) are two events satisfying \(C \subset D\) with \(P(D)\) \(\neq 0\), then the correct statement among the following is [2011] (A) \(P(C \mid D) \geq P(C)\) (B) \(P(C \mid D)

Three numbers are chosen at random without replacement from first eight natural numbers. The probability that their minimum is 3 , given that their maximum is 6, is (A) \(\frac{3}{8}\) (B) \(\frac{1}{5}\) (C) \(\frac{1}{4}\) (D) \(\frac{2}{5}\)

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is [2013] (A) \(\frac{13}{3^{5}}\) (B) \(\frac{11}{3^{5}}\) (C) \(\frac{10}{3^{5}}\) (D) \(\frac{17}{3^{5}}\)

Let \(\mathrm{A}\) and \(\mathrm{B}\) be two events such that \(p(\overline{A \cup B})=\frac{1}{6}, p(A \cap B)=\frac{1}{4}\) and \(p(\bar{A})=\frac{1}{4}\), where \(\bar{A}\) stands for the complement of the event \(A\). Then, the events \(A\) and \(B\) are (A) mutually exclusive and independent (B) equally likely but not independent (C) independent but not equally likely (D) independent and equally likely

Let two fair six-faced dice \(\mathrm{A}\) and \(\mathrm{B}\) be thrown simultaneously If \(\mathrm{E}\), is the event that die A shows up four, \(\mathrm{E} 2\) is the event that die \(\mathrm{B}\) shows up two and \(\mathrm{E} 3\) is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true? (A) E, E2 and E3 are independent. (B) E, and E2 are independent. (C) E2 and E3 are independent. (D) E, and E3 are independent.