Probability PYQs Set 3 (PDF)

Set 3 of 16 · Questions 61-90 of 458

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 3 (Q61-90)
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61. $Let a random variable X take values 0, 1, 2, 3 with P (X = 0) = P (X = 1) = p, P (X = 2) = P (X = 3) and E (X^{2}) = 2 E (X) . Then the value of 8 p - 1 is:$ (1) $2$ (2) $0$ (3) $3$ (4) $1$ 62. $Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99 % is:$ (1) $5$ (2) $6$ (3) $8$ (4) $7$ 63. $A fair coin is tossed a fixed number of times. If the probability of getting five heads is equal to that of getting seven heads, then the probability of getting four heads is$ (1) $\frac{495}{4096}$ (2) $\frac{429}{2048}$ (3) $\frac{165}{1024}$ (4) $\frac{35}{512}$ 64. $In a game, a man wins ₹ 100 if he gets 5 or 6 on a throw of a fair die and loses ₹ 50 for getting any other number on the die. If he decides to throw the die either till he gets a five ora six or to a maximum of three throws, then his expected gain/loss (in rupees) is:$ (1) $\frac{400}{9} loss$ (2) $0$ (3) $\frac{400}{3} gain$ (4) $\frac{400}{3} loss$ 65. $An unbiased coin is tossed. If the outcome is a headthen a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1, 2, 3, \dots, 9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is:$ (1) $\frac{13}{36}$ (2) $\frac{15}{72}$ (3) $\frac{19}{72}$ (4) $\frac{19}{36}$ 66. $There are three bags X, Y and Z . Bag X contains 5 one-rupee coins and 4 five-rupee coins; Bag Y contains 4 one-rupee coins and 5 five-rupee coins and Bag Z contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y , is :$ (1) $\frac{1}{4}$ (2) $\frac{1}{2}$ (3) $\frac{5}{12}$ (4) $\frac{1}{3}$ 67. $A pair of dice is thrown. Then the probability that either of the dice shows 2 when their sum is 6 is$ (1) $\frac{1}{2}$ (2) $\frac{1}{5}$ (3) $\frac{2}{5}$ (4) $\frac{3}{5}$ 68. $If \frac{1 + 3 P}{3}, \frac{1 - 2 P}{2} are probabilities of two mutually exclusive events, then P lies in the interval.$ (1) $(- \frac{1}{3}, \frac{1}{2})$ (2) $(\frac{- 1}{2}; \frac{1}{2})$ (3) $(- \frac{1}{3}, \frac{2}{3})$ (4) $(\frac{- 1}{3}, \frac{2}{3})$ 69. $If two subsets A and B are selected at random from a set S containing n elements, then the probability that A \cap B = φ and A \cup B = S, is$ (1) $\frac{1}{2 ^{n}}$ (2) $2^{n}$ (3) $1 2^{n} +1$ (4) $\frac{1}{2 ^{n} \times 2 ^{n}}$ 70. $A bag contains 3 white and 6 red balls. Four balls are drawn at a time randomly. Then the probability of getting at least two red balls is$ (1) $\frac{8}{27}$ (2) $\frac{5}{14}$ (3) $\frac{60}{63}$ (4) $\frac{1}{2}$ 71. $One out of 9 ships is likely to sink, when they are set on sail. When 6 ships are set on sail, the probability that exactly 3 of them will not arrive safely is$ (1) $1 - \frac{1}{9 ^{6}}$ (2) $^{6} C_{3} \frac{8 ^{3}}{9 ^{6}}$ (3) $\frac{25 \times 8 ^{3}}{9 ^{5}}$ (4) $^{6} C_{3} \frac{8}{9 ^{6}}$ 72. $Two bad eggs are mixed accidentally with 10 good ones. If three eggs are drawn at random from this lot in succession without replacement, then the variance of the probability distribution of the number of bad eggs drawn is$ (1) $\frac{17}{44}$ (2) $\frac{15}{44}$ (3) $\frac{13}{44}$ (4) $\frac{9}{44}$ 73. $Let X and Y be two events such that P(X \| Y) = \frac{1}{2}, P(Y \| X) = \frac{1}{3} and P(X \cap Y) = \frac{1}{6} . Which of the following is (are) correct ?$ (1) $P(X \cup Y) = \frac{2}{3}$ (2) $X and Y are independent$ (3) $X and Y are not independent$ (4) $P(XC \cap Y) = \frac{1}{3}$ 74. $An integer is chosen from {2 k / - 9 \leq k \leq 10} . The probability that it is divisible by both 4 and 6 is$ (1) $\frac{1}{10}$ (2) $\frac{1}{20}$ (3) $\frac{1}{4}$ (4) $\frac{3}{20}$ 75. $Let A and B be two non-null events such that A \subset B . Then, which of the following statements is always correct?$ (1) $P (A ∣ B) = P (B) - P (A)$ (2) $P(A∣B)\geqP(A)$ (3) $P(A∣B)\leqP(A)$ (4) $P (A ∣ B) = 1$ 76. $A, B, C are three events, one of which must and only one can happen. The odds in favour of A are 4 : 6, the odds against B are 7 : 3 . Thus, odds against C are$ (1) $7 : 3$ (2) $4 : 6$ (3) $6 : 4$ (4) $3 : 7$ 77. $If the probability that the random variable X takes the value x is given by P (X = x) = k (x + 1) 3^{- x}, x = 0, 1, 2, 3 \dots, where k is a constant, then P (X \geq 3) is equal to$ (1) $\frac{1}{9}$ (2) $\frac{8}{27}$ (3) $\frac{7}{27}$ (4) $\frac{4}{9}$ 78. $The random variable X has Binomial distribution B (20, 0.4) . Then find 5 - 5 P (X \geq 2) =$ (1) $62 (\frac{2}{5})^{19}$ (2) $43 (\frac{3}{5})^{19}$ (3) $1 + 23 (\frac{3 ^{19}}{5 ^{20}})$ (4) $1 + 62 (\frac{2 ^{19}}{5 ^{20}})$ 79. $Sum of mean and variance of a binomial distribution of 6 trials is \frac{270}{49} . Find the probability of success.$ (1) $5/7$ (2) $3/5$ (3) $7/9$ (4) $2/11$ 80. $In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is:$ (1) $\frac{5}{31}$ (2) $\frac{31}{61}$ (3) $\frac{5}{6}$ (4) $\frac{30}{61}$ 81. $The p.d.f. of a c.r.v. X is f (x) = {6 x (1 - x), 0, 0 \leq x \leq 1 elsewhere If P (X < a) = P (X > a), then a =$ (1) $1$ (2) $\frac{1}{2}$ (3) $\frac{1}{3}$ (4) $\frac{1}{4}$ 82. $If three dice are thrown then the probability that the sum of the numbers on their uppermost faces to be at least 5 is$ (1) $1/53$ (2) $53/54$ (3) $1/54$ (4) $52/53$ 83. $In a lottery, containing 35 tickets, exactly 10 tickets bear a prize. If a ticket is drawn at random, then the probability of not getting a prize is.$ (1) $\frac{1}{10}$ (2) $\frac{2}{5}$ (3) $\frac{2}{7}$ (4) $\frac{5}{7}$ 84. $A pandemic has been spreading all over the world. The probabilities are 0.7 that there will be a lockdown, 0.8 that the pandemic is controlled in one month if there is a lockdown and 0.3 that it is controlled in one month if there is no lockdown. The probability that the pandemic will be controlled in one month is$ (1) $0.65$ (2) $1.46$ (3) $1.65$ (4) $0.46$ 85. $If the mean and variance of a binomial variable X are 2.4 and 1.44 respectively, then the parameters n and p are respectively$ (1) $6, \frac{2}{5} .$ (2) $4, \frac{3}{5}$ (3) $6, \frac{3}{5}$ (4) $8, \frac{1}{3}$ 86. $Let X \sim B (n, p) with mean μ and variance σ^{2} . If μ = 2 σ^{2} and μ + σ^{2} = 3, then P (X \leq 3$ (1) $\frac{15}{16}$ (2) $\frac{2}{3}$ (3) $\frac{14}{17}$ (4) $\frac{1}{3}$ 87. $If X is a random variable such that P (X = - 2) = P (X = - 1) = P (X = 2) = P (X = 1) = \frac{1}{6} and P (X = 0) = \frac{1}{3}, then the mean of X is$ (1) $\frac{5}{3}$ (2) $1$ (3) $0$ (4) $\frac{3}{5}$ 88. $Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is:$ (1) $\frac{5}{18}$ (2) $\frac{5}{16}$ (3) $\frac{4}{17}$ (4) $\frac{7}{18}$ 89. $If A and B are events of a sample space such that P (A \cup B) = \frac{3}{4}, P (A \cap B) = \frac{1}{4} and P (A ˉ) = \frac{2}{3}, then P (A ˉ \cap B) is$ (1) $\frac{5}{12}$ (2) $\frac{3}{8}$ (3) $\frac{4}{5}$ (4) $\frac{5}{4}$ 90. $If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is p, then 96 p is equal to_____.$