Chapter 15

Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is: \(\quad\) [Sep. 06, 2020 (I)] (a) \(\frac{15}{101}\) (b) \(\frac{5}{101}\) (c) \(\frac{5}{33}\) (d) \(\frac{10}{99}\)

If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is : \(\quad\) [Jan. 9,2020 (II)] (a) \(\frac{965}{2^{11}}\) (b) \(\frac{965}{2^{10}}\) (c) \(\frac{945}{2^{10}}\) (d) \(\frac{945}{2^{11}}\)

Let \(\mathrm{S}=\\{1,2, \ldots ., 20\\}\). A subset \(\mathrm{B}\) of \(\mathrm{S}\) is said to be "nice", if the sum of the elements of \(\mathrm{B}\) is 203 . Than the probability that a randomly chosen subset of \(\mathrm{S}\) is "nice" is : (a) \(\frac{7}{2^{20}}\) (b) \(\frac{5}{2^{20}}\) (c) \(\frac{4}{2^{20}}\) (d) \(\frac{6}{2^{20}}\)

Two different families \(A\) and \(B\) are blessed with equal number of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family \(B\) is \(\frac{1}{12}\), then the number of children in each family is? [Online April 16, 2018] (a) 4 (b) 6 (c) 3 (d) 5

A box \(A^{\prime}\) contanis 2 white, 3 red and 2 black balls. Another box ' \(B\) ' contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box \({ }^{\prime} B^{\prime}\) is [Online April 15, 2018] (a) \(\frac{7}{16}\) (b) \(\frac{9}{32}\) (c) \(\frac{7}{8}\) (d) \(\frac{9}{16}\)

A number \(x\) is chosen at random from the set \(\\{1,2,3,4, \ldots .\), 100\\}. Define the event: \(\mathrm{A}=\) the chosen number \(\mathrm{x}\) satisfies \(\frac{(x-10)(x-50)}{(x-30)} \geq 0\) Then \(\mathrm{P}(\mathrm{A})\) is: \(\quad[\) Online April 12, 2014] (a) \(0.71\) (b) \(0.70\) (c) \(0.51\) (d) \(0.20\)

A set \(S\) contains 7 elements. A non-empty subset \(A\) of \(S\) and an element \(\mathrm{x}\) of \(\mathrm{S}\) are chosen at random. Then the probability that \(x \in A\) is: \(\quad\) Online April 11, 2014] (a) \(\frac{1}{2}\) (b) \(\frac{64}{127}\) (c) \(\frac{63}{128}\) (d) \(\frac{31}{128}\)

If six students, including two particular students \(A\) and \(B\), stand in a row, then the probability that \(A\) and \(B\) are separated with one student in between them is [Online May 19, 2012] (a) \(\frac{8}{15}\) (b) \(\frac{4}{15}\) (c) \(\frac{2}{15}\) (d) \(\frac{1}{15}\)

A number \(n\) is randomly selected from the set \(\\{1,2,3, \ldots ., 1000\\} .\) The probability that \(\frac{\sum_{i=1}^{n} i^{2}}{\sum_{i=1}^{n} i}\) is an integer is \(\quad\) [Online May 12, 2012] (a) 0331 (b) \(0.333\) (c) 0334 (d) \(0.332\)

Four numbers are chosen at random (without replacement) from the set \(\\{1,2,3, \ldots 20\\}\) [2010] Statement -1: The probability that the chosen numbers when arranged in some order will form an AP is \(\frac{1}{85}\). Statement \(-2:\) If the four chosen numbers form an AP, then the set of all possible values of common difference is \((\pm 1, \pm 2, \pm 3, \pm 4, \pm 5)\) (a) Statement \(-1\) is true, Statement \(-2\) is true; Statement \(-2\) is not a correct explanation for Statement \(-1\) (b) Statement \(-1\) is true, Statment \(-2\) is false (c) Statement \(-1\) is false, Statment \(-2\) is true. (d) Statement \(-1\) is true, Statement \(-2\) is true; Statement 2 is a correct explanation for Statement \(-1\).

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is (a) \(\frac{2}{7}\) (b) \(\frac{1}{21}\) (c) \(\frac{2}{23}\) (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) \(\frac{2}{5}\) (b) \(\frac{4}{5}\) (c) \(\frac{3}{5}\) (d) \(\frac{1}{5}\)

The probabilities of three events \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are given by \(\mathrm{P}(\mathrm{A})=0.6, \mathrm{P}(\mathrm{B})=0.4\) and \(\mathrm{P}(\mathrm{C})=0.5\). If \(\mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.8, \mathrm{P}\) \((\mathrm{A} \cap \mathrm{C})=0.3, \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0.2, \mathrm{P}(\mathrm{B} \cap \mathrm{C})=\beta\) and \(\mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=\alpha\), where \(0.85 \leq \alpha \leq 0.95\), then \(\beta\) lies in the interval: \(\quad\) [Sep.06, 2020 (II)] (a) \([0.35,0.36]\) (b) \([0.25,0.35]\) (c) \([0.20,0.25]\) (d) \([0.36,0.40]\)

Let \(A\) and \(B\) be two events such that the probability that exactly one of them occurs is \(\frac{2}{5}\) and the probability that \(A\) or \(B\) occurs is \(\frac{1}{2}\), then the probability of both of them occur together is: \(\quad\) [Jan. 8, 2020 (II)] (a) \(0.02\) (b) \(0.20\) (c) \(0.01\) (d) \(0.10\)

In a class of 60 students, 40 opted for \(\mathrm{NCC}, 30\) opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCCnor for NSS is: [Jan. 12, 2019 (II)] (a) \(\frac{1}{6}\) (b) \(\frac{1}{3}\) (c) \(\frac{2}{3}\) (d) \(\frac{5}{6}\)

For three events \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\), \(\mathrm{P}\) (Exactly one of \(\mathrm{A}\) or \(\mathrm{B}\) occurs) \(=\mathrm{P}(\) Exactly one of \(\mathrm{B}\) or \(\mathrm{C}\) occurs \()\) \(=\mathrm{P}\) (Exactly one of \(\mathrm{C}\) or A occurs) \(=\frac{1}{4}\) and \(\mathrm{P}\) (All the three events occur simultaneously) \(=\frac{1}{16}\). Then the probability that at least one of the events occurs, is : (a) \(\frac{3}{16}\) (b) \(\frac{7}{32}\) (c) \(\frac{7}{16}\) (d) \(\frac{7}{64}\)

From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for these committees to have more women than men, is : (a) \(\frac{21}{220}\) (b) \(\frac{3}{11}\) (c) \(\frac{1}{11}\) (d) \(\frac{2}{23}\)

If \(\mathrm{A}\) and \(\mathrm{B}\) are two events such that \(\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A} \cap \mathrm{B})\), then the incorrect statement amongst the following statements is: [Online April 9, 2014] (a) \(\mathrm{A}\) and \(\mathrm{B}\) are equally likely (b) \(\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{r}\right)=0\) (c) \(\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)=0\) (d) \(\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=1\)

If the events \(\mathrm{A}\) and \(\mathrm{B}\) are mutually exclusive events such that \(\mathrm{P}(\mathrm{A})=\frac{3 x+1}{3}\) and \(\mathrm{P}(\mathrm{B})=\frac{1-x}{4}\), then the set of possible values of \(x\) lies in the interval : \mathrm{\\{} O n l i n e ~ A p r i l ~ 2 5 , ~ 2 0 1 3 ] ~ (a) \([0,1]\) (b) \(\left[\frac{1}{3}, \frac{2}{3}\right]\) (c) \(\left[-\frac{1}{3}, \frac{5}{9}\right]\) (d) \(\left[-\frac{7}{9}, \frac{4}{9}\right]\)

Let \(X\) and \(Y\) are two events such that \(P(X \cup Y)=P(X \cap Y)\) Statement 1: \(P\left(X \cap Y^{\prime}\right)=P\left(X^{\prime} \cap Y\right)=0\) Statement 2: \(P(X)+P(Y)=2 P(X \cap Y)\) [Online May 7, 2012] (a) Statement 1 is false, Statement 2 is true. (b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 . (c) Statement 1 is true, Statement 2 is false. (d) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1 .

A die is thrown. Let \(A\) be the event that the number obtained is greater than \(3 .\) Let \(B\) 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) \(\frac{2}{5}\)

Events \(A, B, C\) are mutually exclusive events such that \(P(A)=\frac{3 x+1}{3}, P(B)=\frac{1-x}{4}\) and \(P(C)=\frac{1-2 x}{2}\) The set of possible values of \(x\) are in the interval. (a) \([0,1]\) (b) \(\left[\frac{1}{3}, \frac{1}{2}\right]\) (c) \(\left[\frac{1}{3}, \frac{2}{3}\right]\) (d) \(\left[\frac{1}{3}, \frac{13}{3}\right]\)

\(A\) and \(B\) are events such that \(P(A \cup B)=3 / 4, P(A \cap B)=1 / 4\), \(P(\bar{A})=2 / 3\) then \(P(\bar{A} \cap B)\) is \(\quad\) [2002] (a) \(5 / 12\) (b) \(3 / 8\) (c) \(5 / 8\) (d) \(1 / 4\)