Probability PYQs Set 15 (PDF)

Set 15 of 16 · Questions 421-450 of 458

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 15 (Q421-450)
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421. $If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is$ (1) $\frac{55}{3} (\frac{2}{3})^{11}$ (2) $55 (\frac{2}{3})^{10}$ (3) $220 (\frac{1}{3})^{12}$ (4) $22 (\frac{1}{3})^{11}$ 422. $A and B are two independent events. P (A) = \frac{2}{5}, P (B) = \frac{1}{3} . Match the following. List-I (A) P (A \cup B) (B) P (\frac{A}{B}) (C) P (A \cup B) (D) P (\frac{B}{A}) List-II (I) \frac{2}{3} (II) \frac{11}{15} (III) \frac{3}{5} (IV) \frac{2}{5} (V) \frac{1}{3} The correct answer is (A, B, C, D) =$ (1) $I, III, IV, II$ (2) $II, IV, V, I$ (3) $II, IV, III, V$ (4) $II, IV, III, I$ 423. $If the p.d.f. of a r.v. X is given as x_{i} -2 -1 012 P (X = x_{i}) 0.2 0.3 0.15 0.25 0.1 then F (0) =$ (1) $P (X < 0)$ (2) $P (X > 0)$ (3) $1 - P (X > 0)$ (4) $1 - P (X < 0)$ 424. $Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :$ (1) $\frac{1}{8}$ (2) $\frac{5}{8}$ (3) $\frac{5}{16}$ (4) $1$ 425. $Let 9 distinct balls be distributed among 4 boxes, B_{1}, B_{3} and B_{4} . If the probability than B_{3} contains exactly 3 balls is k (\frac{3}{4})^{9} then k lies in the set :$ (1) ${x \inR : \|x - 3\| < 1}$ (2) ${x \in R : \|x - 2\| \leq 1}$ (3) ${x \inR : \|x - 1\| < 1}$ (4) ${x \in R : \|x - 5\| \leq 1}$ 426. $Let E_{1} and E_{2} be two events such that the conditional probabilities P(E_{1} \| E_{2}) = \frac{1}{2}, P(E_{2} \| E_{1})= \frac{3}{4} and P(E_{1} \cap E_{2}) = \frac{1}{8} . Then:$ (1) $P (E_{1} \cap E_{2}) = P (E_{1}) \cdot P (E_{2})$ (2) $P (E_{1}^{'} \cap E_{2}^{'}) = P (E_{1}^{'}) \cdot P (E_{2})$ (3) $P_{(} E_{1} \cap E_{2}^{^{'}}_{)} = P_{(} E_{1)} . P_{(} E_{2)}$ (4) $P_{(} E_{1}^{^{'}} \cap E_{2)} = P_{(} E_{1)} . P_{(} E_{2)}$ 427. $Let be a random variable such that the probability function of a distribution is given by P (X = 0) = \frac{1}{2}, P (X = j) = \frac{1}{3 ^{j}} (j = 1, 2, 3, ..., \infty) . Then the mean of the distribution and P(X is positive and even) respectively are:$ (1) $\frac{3}{8} and \frac{1}{8}$ (2) $\frac{3}{4} and \frac{1}{8}$ (3) $\frac{3}{4} and \frac{1}{9}$ (4) $\frac{3}{4} and \frac{1}{16}$ 428. $A random variable X has the following probability distribution : X 01234 P(X) k 2k 4k 6k 86 The value of P(l < X < 4 \|X \leq 2) is equal to :$ (1) $\frac{4}{7}$ (2) $\frac{2}{3}$ (3) $\frac{3}{7}$ (4) $\frac{4}{5}$ 429. $In a college, 40 % students attend Mathematics class, 30 % students attend physics class and 20 % students attend both the classes. If a student is chosen at random from the college, the probability that the student chosen attend only one class is$ (1) $\frac{3}{10}$ (2) $\frac{1}{3}$ (3) $\frac{2}{3}$ (4) $\frac{1}{4}$ 430. $Two numbers are selected at random from the set {1, 2, 3, \dots . 13} . If the sum of the selected numbers is even, the probability that both the numbers are odd is$ (1) $\frac{2}{13}$ (2) $\frac{1}{2}$ (3) $\frac{7}{12}$ (4) $\frac{5}{26}$ 431. $If A and B are two events such that P(a)= \frac{1}{3},P(b) = \frac{1}{5} and P(A \cup B) = \frac{1}{2}, then P(A\|B') + P(B\|A') is equal to$ (1) $\frac{3}{4}$ (2) $\frac{5}{8}$ (3) $\frac{5}{4}$ (4) $\frac{7}{8}$ 432. $If two unbiased dice are rolled simultaneously until a sum of the number appeared on these dice is either 7 or 11 , then the probability that 7 comes before 11 , is$ (1) $\frac{3}{8}$ (2) $\frac{3}{4}$ (3) $\frac{5}{6}$ (4) $\frac{2}{9}$ 433. $A five digit number is formed by the digits 1, 2, 3, 4, 5 with no digit being repeated. The probability that the number is divisible by 4 , is$ (1) $\frac{1}{5}$ (2) $\frac{2}{5}$ (3) $\frac{3}{5}$ (4) $\frac{4}{5}$ 434. $A, B and C are aiming to shoot a balloon. A will succeed 4 times out of 6 attempts. The chance of B to shoot the balloon is 3 out of 5 and that of C is 2 out of 3 . If the three aim to shoot the balloon simultaneously. Then, the probability that at least two of them hit the balloon is$ (1) $\frac{5}{9}$ (2) $\frac{9}{25}$ (3) $\frac{4}{9}$ (4) $\frac{32}{45}$ 435. $Two integers are drawn at random from the set {5, 6, .., \dots 35} . What is the probability that their difference is odd?$ (1) $\frac{15}{62}$ (2) $\frac{8}{31}$ (3) $\frac{15}{31}$ (4) $\frac{16}{31}$ 436. $Suppose that a bag A contains n red and 2 black balls and another bag B contains 2 red and n black balls. One of the two bags is selected at random and two balls are drawn from it at a time. When it is known that the two balls drawn are red, if the probability that those two balls drawn are from bag A is \frac{6}{7}, then n =$ (1) $6$ (2) $4$ (3) $8$ (4) $7$ 437. $A bag contains 4 red and 3 black balls. One ball is drawn and then replaced in the bag and the process is repeated. Let X denote the number of times black ball is drawn in 3 draws. Assuming that at each draw each ball is equally likely to be selected, then the probability distribution of X is given by$ (1) $X 0123 P(X) (\frac{3}{7})^{3} \frac{12}{7} (\frac{3}{7})^{2} \frac{9}{7} (\frac{4}{7})^{2} (\frac{4}{7})^{3}$ (2) $X 0123 P(X) (\frac{4}{7})^{3} \frac{9}{7} (\frac{4}{7})^{2} \frac{12}{7} (\frac{3}{7})^{2} (\frac{3}{7})^{3}$ (3) $X 0123 P(X) (\frac{4}{7})^{3} \frac{12}{7} (\frac{4}{7})^{2} \frac{9}{7} (\frac{3}{7})^{2} (\frac{3}{7})^{3}$ (4) $X 123 P (X) \frac{1}{3} \frac{1}{3} \frac{1}{3}$ 438. $ω is a complex cube root of unity. When an unbiased die is thrown 3 times, if β_{1}, β_{2} and β_{3} are the numbers appeared on the die, then the probability that β_{1}, β_{2} and β_{3} satisfy ω^{β_{1}} + ω^{β_{2}} = - ω^{β_{3}} is$ (1) $\frac{212}{513}$ (2) $\frac{1}{3}$ (3) $\frac{3}{5}$ (4) $\frac{2}{9}$ 439. $A bag P contains 5 white marbles and 3 black marbles. Four marbles are drawn at random from P and are put in an empty bag Q . If a marble drawn at random from Q is found to be black then the probability that all the three black marbles in P are transferred to the bag Q is.$ (1) $\frac{1}{7}$ (2) $\frac{6}{7}$ (3) $\frac{1}{8}$ (4) $\frac{7}{8}$ 440. $There are 10 red and 5 yellow roses of different sizes. If x is the number of garlands that can be formed with all these flowers so that no two yellow roses come together and y is the number of garlands formed with all these flowers so that all the red roses coming together, then \frac{2 ( x - y )}{10 !} =$ (1) $\frac{9 !}{5 !} - 5!$ (2) $(11)^{2} . (4!)$ (3) $10! - 6!$ (4) $6! \times (5! - 2)$ 441. $In a battery manufacturing factory, machines P, Q and R manufacture 20 %, 30 % and 50 % respectively of the total output. The chances that a defective battery is produced by these machines are 1 %, 1.5 % and 2 % respectively. If a battery is selected as random from production, then the probability that it is defective is$ (1) $\frac{69}{2000}$ (2) $\frac{33}{2000}$ (3) $\frac{1}{40}$ (4) $\frac{29}{2000}$ 442. $If a proper divisor of the integer 2520 is selected at random, then the probability that it is an odd number is$ (1) $\frac{11}{46}$ (2) $\frac{12}{46}$ (3) $\frac{11}{48}$ (4) $\frac{1}{4}$ 443. $For the probability distribution given by X = x_{i} 012 P_{i} \frac{25}{36} \frac{5}{18} \frac{1}{36} the standard deviation (σ) is$ (1) $\frac{1}{3}$ (2) $\frac{1}{3} \frac{5}{2}$ (3) $\frac{5}{36}$ (4) $None of the above$ 444. $Two cards are drawn from a pack of 52 playing cards one after the other. If p_{1} is the probability of getting a queen in the first draw and a diamond card in the second draw when the first card drawn is replaced and p_{2} is the probability of the same event when the first card drawn is not replaced. Then \frac{p _{1}}{p _{2}} =$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 445. $Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability 0.7 where as George, independently, hits the target with probability 0.4. Given that exactly one shot hit the target, what is the probability that it was George's shot?$ (1) $\frac{2}{3}$ (2) $\frac{2}{9}$ (3) $\frac{1}{9}$ (4) $\frac{8}{9}$ 446. $Two persons P and Q are considering to apply for a job. The probability that P applies for the job is 1/4, the probability that P applies for the job given that Q applies for the job is 1/2, and the probability that Q applies for the job given that P applies for the job is 1/3 . Then the probability that P does not apply for the job given that Q does not apply for the job is$ (1) $\frac{4}{5}$ (2) $\frac{5}{6}$ (3) $\frac{7}{8}$ (4) $\frac{11}{12}$ 447. $Probability that a person will develop immunity after vaccination is 0.8 . If 8 people are given the vaccine then probability that all develop immunity is$ (1) $(0.2)^{8}$ (2) $(0.8)^{8}$ (3) $1$ (4) $^{8} C_{6} (0.2)^{6} (0.8)^{2}$ 448. $Let A and B be not mutually exclusive events. If P (A) = \frac{4}{9}, P (A \cap B) = \frac{3}{7} then P (\frac{B ˉ}{A}) =$ (1) $0$ (2) $\frac{1}{28}$ (3) $\frac{3}{13}$ (4) $\frac{4}{7}$ 449. $In a family with 4 children, the probability that there are at least two girls, is$ (1) $\frac{1}{2}$ (2) $\frac{9}{16}$ (3) $\frac{3}{4}$ (4) $\frac{11}{16}$ 450. $A fair coin is tossed n times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is ___$