Probability PYQs Set 2 (PDF)

Set 2 of 16 · Questions 31-60 of 458

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 2 (Q31-60)
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31. $A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is:$ (1) $1 2 gain$ (2) $\frac{1}{4} loss$ (3) $\frac{1}{2} loss$ (4) $2 gain$ 32. $Two urns identical in appearance contain respectively 3 green and 2 black balls and 2 green and 5 black balls. One urn is selected at random and a ball is drawn from it. The probability that it is black is$ (1) $\frac{39}{70}$ (2) $\frac{37}{70}$ (3) $\frac{41}{70}$ (4) $\frac{33}{70}$ 33. $The probability that an event does not happen in one trial is 0.8 The probability that the event happens at most once in three trials is$ (1) $0.896$ (2) $0.791$ (3) $0.642$ (4) $0.592$ 34. $In a Binomial distribution, n = 4 and 2 P (X = 3) = 3 P (X = 2), then q =$ (1) $\frac{4}{13}$ (2) $\frac{2}{13}$ (3) $\frac{11}{13}$ (4) $\frac{9}{13}$ 35. $The probability of solving a problem by three persons A, B and C independently is \frac{1}{2}, \frac{1}{4} and \frac{1}{3} respectively. Then the probability of the problem is solved by any two of them is$ (1) $\frac{1}{4}$ (2) $\frac{1}{24}$ (3) $\frac{1}{8}$ (4) $\frac{1}{12}$ 36. $In a game, two dice are thrown simultaneously by a person A and two cards are drawn at random simultaneously from a pack of 52 playing cards by a person B. They win the game if A gets a prime score as the sum of the numbers appear on both the dice and B gets a face card and a card having a prime number. Then the probability that both A and B win is$ (1) $\frac{8}{663}$ (2) $\frac{40}{663}$ (3) $\frac{16}{117}$ (4) $\frac{40}{221}$ 37. $Let p (x) represent the probability mass function of a poisson distribution. If its mean λ = 3.725, then value of x at which p (x) is maximum is$ (1) $2$ (2) $3$ (3) $4$ (4) $5$ 38. $If a random variable X takes the values x_{1}, x_{2}, x_{3}, \dots, x_{100} with probability P (X = x_{i}) = Ki (i + 1), then 200 K =$ (1) $\frac{1}{1707}$ (2) $\frac{1}{1717}$ (3) $\frac{1}{1727}$ (4) $\frac{1}{1777}$ 39. $Two dice are thrown simultaneously. The probability of getting two numbers whose product is even is$ (1) $1/2$ (2) $3/4$ (3) $3/8$ (4) $5/16$ 40. $Let A = (a_{ij}) be a 2 \times 2 matrix such that a_{ij} \in 0, 1 for all i and j . Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of X is:$ (1) $\frac{3}{4}$ (2) $\frac{5}{8}$ (3) $\frac{3}{8}$ (4) $\frac{1}{4}$ 41. $Two friends A and B meet every weekend either at a party or at a sports club. The probability that they meet at sports club is \frac{4}{9} . The probability that they will dine together at a party and at the club are respectively \frac{1}{3} and \frac{2}{5} . On a certain weekend the probability that they disperse without dine together is$ (1) $\frac{86}{135}$ (2) $\frac{10}{27}$ (3) $\frac{17}{27}$ (4) $\frac{56}{135}$ 42. $A man is known to speak the truth 2 out of 3 times. If he throws a die and reports that it is six, then the probability that it is actually five, is$ (1) $\frac{3}{8}$ (2) $\frac{1}{7}$ (3) $\frac{2}{7}$ (4) $\frac{4}{5}$ 43. $Person A can solve 90 % of the problems given in the book and Person B can solve 70 % . Then the probability that atleast one of them will solve the problem selected at random from the book is$ (1) $0.16$ (2) $0.69$ (3) $0.97$ (4) $0.20$ 44. $If the coefficients a and b of a quadratic expression x^{2} + ax + b are choosen from the sets A = 3, 4, 5 and B = 1, 2, 3, 4 respectively, then the probability that the equation x^{2} + ax + b = 0 has real roots is$ (1) $\frac{1}{6}$ (2) $\frac{5}{6}$ (3) $\frac{3}{4}$ (4) $\frac{7}{12}$ 45. $A box contains 30 toys of same size in which 10 toys are white and all the remaining toys are blue. A toy is drawn at random from the box and it is replaced in the box after noting down its colour. If 5 toys are drawn in this way, then the probability of getting at most 2 white toys is$ (1) $(\frac{6}{9})^{2}$ (2) $(\frac{8}{9})^{2}$ (3) $(\frac{7}{9})^{2}$ (4) $(\frac{2}{3})^{5}$ 46. $If P (A \cap B) = \frac{7}{10} and P (B) = \frac{17}{20}, where P stands for probability then P (A / B) is equal to$ (1) $\frac{7}{8}$ (2) $\frac{17}{20}$ (3) $\frac{14}{17}$ (4) $\frac{1}{8}$ 47. $For the following distribution function F (X) of ar.y. X X 123456 F (X) 0.2 0.37 0.48 0.62 0.85 1 P (3 < X \leq 5) =$ (1) $0.48$ (2) $0.37$ (3) $0.27$ (4) $1.47$ 48. $Let X be a random variable such that X (x) is the number of heads is X for each x \in S, where S is the sample space of random experiment of tossing three fair coins simultaneously. Find the value of ρ (X^{- 1} (2)) .$ (1) $\frac{3}{8}$ (2) $\frac{5}{8}$ (3) $\frac{1}{8}$ (4) $\frac{3}{4}$ 49. $The p.d.f. of a random variable x is given by f (x) = {\frac{1}{4 a}, 0, 0 < x < 4 a (a > 0) otherwise and P (x < \frac{3 a}{2}) = k P (x > \frac{5 a}{2}) then k =$ (1) $1$ (2) $1/4$ (3) $1/8$ (4) $1/2$ 50. $A box contains 10 mangoes out of which 4 are spoiled. 2 mangoes are taken together at random. If one of them is found to be good, then the probability that the other is also good, is$ (1) $\frac{1}{3}$ (2) $\frac{2}{3}$ (3) $\frac{8}{15}$ (4) $\frac{5}{13}$ 51. $If A and B are two events of a random experiment such that P (A \cup B) = P (A \cap B), then which one amongst the following four options in not true?$ (1) $A and B are equally likely$ (2) $P (A \cap B^{’}) = 0$ (3) $P (A^{’} \cap B) = 0$ (4) $P (A) + P (B) = 1$ 52. $Two cards are drawn successively with replacement from well shuffled pack of 52 cards, then the probability distribution of number of queens is$ (1) $X=x 012 P [X = x ∣ \frac{144}{169} \frac{24}{169} \frac{1}{169}$ (2) $X=x 012 P[X=x] \frac{1}{169} \frac{24}{169} \frac{144}{169}$ (3) $X=x 012 P[X=x] \frac{24}{169} \frac{1}{169} \frac{144}{169}$ (4) $X=x 012 P[X=x] \frac{1}{169} \frac{25}{169} \frac{143}{169}$ 53. $Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability ofoccurrence of 0 at even places be \frac{1}{2} and probability of occurrence of 0 at the odd place be \frac{1}{3} . Then theprobability that '10' is followed by '01' is equal to:$ (1) $\frac{1}{18}$ (2) $\frac{1}{3}$ (3) $\frac{1}{6}$ (4) $\frac{1}{9}$ 54. $Three dice are rolled. If the probability of getting different numbers on the three dice is \frac{p}{q}, where p and q are co-prime, then q - p is equal to$ (1) $2$ (2) $1$ (3) $3$ (4) $4$ 55. $Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space S = (x ϵ Z : x (66 - x) \geq \frac{5}{9} M) and the event A=x ϵ S:x is a multiple of 3 Then P(A) is equal to$ (1) $\frac{15}{44}$ (2) $\frac{1}{3}$ (3) $\frac{1}{5}$ (4) $\frac{7}{22}$ 56. $A bag contain 6 white and 4 black balls. Two balls are drawn at random. The probability that they are of the same colour is$ (1) $5/7$ (2) $1/7$ (3) $7/15$ (4) $1/15$ 57. $If A and B are two events such that P (A) = \frac{1}{3}, P (B) = \frac{1}{2} and P (A \cap B) = \frac{1}{6}, then P (\frac{A ^{’}}{B}) is$ (1) $\frac{1}{3}$ (2) $\frac{1}{2}$ (3) $\frac{1}{12}$ (4) $\frac{2}{3}$ 58. $If X is a Poisson random variate with mean 3 , then P (\| X - 3 \| < 2) =$ (1) $\frac{9}{2 e ^{3}}$ (2) $\frac{99}{8 e ^{3}}$ (3) $\frac{3}{2 e ^{3}}$ (4) $\frac{1}{3 e ^{3}}$ 59. $Let the mean and the standard deviation of the probability distribution X α 10 -3 P (X) \frac{1}{3} K \frac{1}{6} \frac{1}{4} be μ and σ, respectively. If σ - μ = 2, then σ + μ is equal to_______$ 60. $An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is p, then 98p is equal to .$