Probability PYQs Set 4 (PDF)

Set 4 of 16 · Questions 91-120 of 458

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 4 (Q91-120)
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91. $A special lottery is to be held to select a student who will live in the only deluxe room available in a hostel. 100 III year, 150 II year and 200 I year students have applied for the room. Each III year student's name is placed in the lottery 3 times, each II year student's name 2 times and I year student's name 1 time. The probability that a III year student gets the room is$ (1) $\frac{1}{8}$ (2) $\frac{2}{9}$ (3) $\frac{2}{7}$ (4) $\frac{3}{8}$ 92. $A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is$ (1) $\frac{5}{6}$ (2) $\frac{1}{6}$ (3) $\frac{5}{11}$ (4) $\frac{6}{11}$ 93. $A bag contains 10 similar balls, of which 4 are blue and 6 are red. Three balls are taken out at random from the bag one after the other without replacement. The probabilities that all the three balls drawn are red is$ (1) $\frac{1}{5}$ (2) $\frac{1}{6}$ (3) $\frac{5}{9}$ (4) $\frac{1}{2}$ 94. $Let X and Y be two events of a sample space such that P (X) = \frac{1}{3}, P (\frac{X}{Y}) = \frac{1}{2} and P (\frac{Y}{X}) = \frac{2}{5} then$ (1) $P (X \cap Y) = \frac{1}{5}$ (2) $P (X \cup Y) = \frac{2}{5}$ (3) $P (Y) = \frac{1}{6}$ (4) $P (\frac{X}{Y}) = \frac{1}{2}$ 95. $In a game, a pair of dice is rolled 24 times. If a person wins the game by not getting 6 on both the dice in any one of the 24 rolls, then the probability that a person wins the game is$ (1) $(\frac{35}{36})^{24}$ (2) $(\frac{17}{18})^{24}$ (3) $(\frac{11}{12})^{24}$ (4) $(\frac{5}{6})^{24}$ 96. $An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is redis:$ (1) $\frac{21}{49}$ (2) $\frac{27}{49}$ (3) $\frac{26}{49}$ (4) $\frac{32}{49}$ 97. $The letters of the word 'QUESTION' are arranged in a row at random. The probability that there are exactly two letters between Q and S is$ (1) $\frac{1}{14}$ (2) $\frac{5}{7}$ (3) $\frac{1}{7}$ (4) $\frac{5}{28}$ 98. $Let A = x, y, z, u and B = a, b . A function f : A \to B is selected randomly. The probability that the function is an onto function is$ (1) $\frac{5}{8}$ (2) $\frac{7}{8}$ (3) $\frac{1}{35}$ (4) $\frac{1}{8}$ 99. $If A and B are events of a random experiment with P (A) = 0.5, P (B) = 0.4 and P (A \cap B) = 0.3, then the probability that neither A nor B occurs is$ (1) $0.04$ (2) $0.4$ (3) $0.8$ (4) $0.2$ 100. $There are 3 bags A, B and C . Bag A contains 2 white and 3 black balls, bag B contains 4 white and 2 black balls and bag C contains 3 white and 2 black balls. If a ball is drawn at random from a randomly chosen bag, then the probability that the ball drawn is black, is$ (1) $\frac{2}{3}$ (2) $\frac{4}{9}$ (3) $\frac{5}{9}$ (4) $\frac{1}{9}$ 101. $Let A and B be two independent events of a random experiment. If the probability that both A and B occur is \frac{1}{6} and the probability that neither of them occur is \frac{1}{3}, then the probability of occurrence of A is$ (1) $0 or 1$ (2) $\frac{1}{2} or \frac{1}{4}$ (3) $\frac{1}{2} or \frac{1}{3}$ (4) $\frac{1}{2} or \frac{1}{7}$ 102. $Two cards are drawn at random from a pack of 52 cards. The probability of these two being "Aces" is$ (1) $\frac{1}{26}$ (2) $\frac{1}{221}$ (3) $\frac{1}{2}$ (4) $\frac{1}{13}$ 103. $A random variable X has the probability distribution X = x_{i} P (X = x_{i}) 1 0.2 2 0.3 3 0.12 4 0.1 5 0.2 6 0.08 If A = {x_{i} : x_{i} is a prime number}, B = {x_{i} : x_{i} < 4} are two events, then P (A \cup B) =$ (1) $0.31$ (2) $0.62$ (3) $0.82$ (4) $0.41$ 104. $A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is \frac{m}{n}, gcd (m, n) = 1, then n^{2} - m^{2} is equal to:$ (1) $64$ (2) $80$ (3) $60$ (4) $72$ 105. $A game consists of tossing a coin 3 times and noting its outcome. A boy wins the game if all tosses give the same outcome (i.e three heads or three tails) and loses the game otherwise. The probability that the boy loses the game is$ (1) $\frac{3}{4}$ (2) $\frac{1}{4}$ (3) $\frac{2}{3}$ (4) $\frac{1}{3}$ 106. $The probability that a mechanic making an error while using a machine on the n th day is given by P (E_{n}) = \frac{1}{2 ^{n}} . If he has operated the machine for 4 days, the probability that he had not made a mistake on 3 of 4 days is$ (1) $\frac{1}{2}$ (2) $\frac{1}{4}$ (3) $\frac{243}{512}$ (4) $\frac{343}{1024}$ 107. $A bag contains 7 green and 5 black balls. 3 balls are drawn at random one after the other. If the balls are not replaced, then the probability of all three balls being green is$ (1) $\frac{343}{1720}$ (2) $\frac{21}{36}$ (3) $\frac{12}{35}$ (4) $\frac{7}{44}$ 108. $The probabilities of three events A, B and C are given by P (A) = 0.6, P (B) = 0.4 and P (C) = 0.5 If P (A \cup B) = 0.8, P (A \cap C) = 0.3, P (A \cap B \cap C) = 0.2, P (B \cap C) = β and P (A \cup B \cup C) = α where 0.85 \leq α \leq 0.95 then β lies in the interval:$ (1) $[0.25, 0.35]$ (2) $[0.35, 0.36]$ (3) $[0.36, 0.40]$ (4) $[0.20, 0.25]$ 109. $When a die is rolled 42 times it is so happened that a face having the digit i occurred 2 i times. The mean deviation from the mean of this discrete frequency distribution is$ (1) $\frac{80}{63}$ (2) $\frac{32}{3}$ (3) $\frac{72}{41}$ (4) $\frac{15}{4}$ 110. $A bag contains 2 n + 1 coins. It is known that n of these coins have head on both sides whereas the other n + 1 coins are fair. One coin is selected at random and tossed. If the probability that toss results in heads is \frac{31}{42}, then the value of n is$ (1) $8$ (2) $5$ (3) $10$ (4) $6$ 111. $If the mean and variance of a binomial variate X are \frac{4}{3}, \frac{8}{9} respectively, then P (X = 2) =$ (1) $\frac{4}{27}$ (2) $\frac{16}{81}$ (3) $\frac{8}{27}$ (4) $\frac{8}{81}$ 112. $A car manufacturing factory has two plants X and Y . Plant X manufactures 70 % of cars and plant Y manufactures 30 % of cars. 80 % of cars at plant X and 90 % of cars at plant Y are rated as standard quality. A car is chosen at random and is found to be standard quality. The probability that it has come from plant X is$ (1) $\frac{56}{73}$ (2) $\frac{56}{84}$ (3) $\frac{56}{83}$ (4) $\frac{56}{79}$ 113. $If A and B are any two events of a sample space, then set-theoretic description for the event: "Exactly one of the events A, B to occur" is (Here E^{c} denotes the compliment of the event E)$ (1) $A \cap B^{c}$ (2) $(A - B) \cup (A \cup B)$ (3) $A \cap B^{c} \cupAc\capB$ (4) $(A \cap B)^{c} \cup (A^{c} \cap B^{c})$ 114. $The coefficients a, b, c in the quadratic equation a x^{2} + bx + c = 0 are chosen from the set 1, 2, 3, 4, 5, 6, 7, 8 . The probability of this equation having repeated roots is :$ (1) $\frac{1}{128}$ (2) $\frac{1}{64}$ (3) $\frac{3}{256}$ (4) $\frac{3}{128}$ 115. $The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90 % is:$ (1) $5$ (2) $3$ (3) $4$ (4) $2$ 116. $For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that thecandidate can solve any problem is \frac{4}{5}, then the probability that he is unable to solve less than two problems is:$ (1) $\frac{201}{5} (\frac{1}{5})^{49}$ (2) $\frac{316}{25} (\frac{4}{5})^{48}$ (3) $\frac{54}{5} (\frac{4}{5})^{49}$ (4) $\frac{164}{25} (\frac{1}{5})^{48}$ 117. $The probability that A wakes up before the alarm rings is 0.4 . Then, the mean and variance of the number of times A wakes up before the alarm rings, in the next 7 days respectively are:$ (1) $0.4, 0.6$ (2) $2.8, 0.6$ (3) $2.8, 1.68$ (4) $7, 0.6$ 118. $A boy tosses fair coin 3 times. If he gets ₹ 2 X for X heads then his expected gain equals to ₹....$ (1) $1$ (2) $\frac{3}{2}$ (3) $3$ (4) $4$ 119. $In a bombing attack, there is 50 % chance that a bomb will hit the target. Atleast two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least 99 % chance of completely destroying the target, is$ 120. $Let x and y be distinct integers where 1 \leq x \leq 25 and 1 \leq y \leq 25 . Then, the number of ways of choosing x and y, such that x + y is divisible by 5. is _______$