Probability PYQs Set 11 (PDF)

Set 11 of 16 · Questions 301-330 of 458

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 11 (Q301-330)
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301. $If f (x) = {3 (1 - 2 x^{2}), 0, 0 < x < 1 otherwise is a probability density function of X, then P (\frac{1}{4} < x < \frac{1}{3}) is$ (1) $\frac{75}{243}$ (2) $\frac{23}{96}$ (3) $\frac{179}{864}$ (4) $\frac{52}{243}$ 302. $A and B are independent events of a random experiment if and only if$ (1) $P (A ∣ B) \neq = P (A \cap B)$ (2) $P (A ∣ B) = P (B ∣ A)$ (3) $P (A ∣ B) \neq = P (A ∣ B^{C})$ (4) $P (A ∣ B) = P (A ∣ B^{C})$ 303. $In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked randomly. The probability that it is neither red nor green is$ (1) $1/3$ (2) $3/4$ (3) $7/19$ (4) $8/21$ 304. $Five digit numbers are formed by using digits 1, 2, 3, 4 and 5 without repetition. Then, the probability that the randomly chosen number is divisible by 4 is$ (1) $\frac{1}{5}$ (2) $\frac{5}{6}$ (3) $\frac{4}{5}$ (4) $\frac{1}{6}$ 305. $If a Poisson variate X satisfies P (X = 2) = P (X = 3), then P (X = 5) =$ (1) $\frac{81}{40 e ^{5}}$ (2) $\frac{81}{40 e ^{3}}$ (3) $\frac{243}{40 e ^{3}}$ (4) $\frac{243}{40 e ^{5}}$ 306. $There are 4 hotels in a town. If 3 men check into the hotels in a day then the probability that each checks into a different hotel is$ (1) $\frac{6}{7}$ (2) $\frac{1}{8}$ (3) $\frac{3}{8}$ (4) $\frac{5}{9}$ 307. $Two integers x and y are chosen with replacement from the set 0, 1, 2, 3, \dots, 10 . Then the probability that \| x - y \| > 5 is :$ (1) $\frac{30}{121}$ (2) $\frac{62}{121}$ (3) $\frac{60}{121}$ (4) $\frac{31}{121}$ 308. $Let two fair six-faced dice A and B be thrown simultaneously. If E1 is the event that die A shows up four, E2 is the event that die B shows up two and E3 is the event that the sum of numbers on both dice is odd, then whic h of the following statements is NOT True ?$ (1) $E2 and E3 are independent$ (2) $E1 and E3 are independent$ (3) $E1, E2 and E3 are independent$ (4) $E1 and E2 are independent$ 309. $Two integers are selected at random from the set 1, 2, \dots, 11 Given that the sum of selected numbers is even, the conditional probability that both the numbers are even is:$ (1) $\frac{7}{10}$ (2) $\frac{1}{2}$ (3) $\frac{2}{5}$ (4) $\frac{3}{5}$ 310. $A r.v. X \sim B (n, p) . If value of mean and variance of X are 18 and 12 respectively, then total number of possible values of X are$ (1) $54$ (2) $55$ (3) $12$ (4) $18$ 311. $There are two dice A and B . Die A has 4 red and 2 white faces and B has 2 red and 4 white faces. A coin is tossed once, if it shows head, die A is rolled, if it shows tail, die B is rolled, if the probability that die A is used is (\frac{32}{33}) when it is given that red turns up every time in first n throws, then n =$ (1) $5$ (2) $6$ (3) $4$ (4) $3$ 312. $If a random variable X has the probability distribution given by P (X = 0) = 3 C^{3}, P (X = 2) = 5 C - 10 C^{2} and P (X = 4) = 4 C - 1, then the variance of that distribution is$ (1) $\frac{68}{9}$ (2) $\frac{22}{9}$ (3) $\frac{612}{81}$ (4) $\frac{128}{81}$ 313. $Three unbiased coins are tossed once. What is the probability of getting at least 2 heads?$ (1) $\frac{1}{7}$ (2) $\frac{1}{3}$ (3) $\frac{1}{5}$ (4) $\frac{1}{2}$ 314. $A flashlight has 10 batteries out of which 4 are dead. If 3 batteries are selected without replacement and tested, then the probability that all 3 are dead is$ (1) $1/30$ (2) $2/8$ (3) $1/15$ (4) $1/10$ 315. $Let S = (1, 2, \dots .., 20) . A subset B of S is said to be "nice", if the sum of the elements of B is 203 . Than the probability that a randomly chosen subset of S is "nice" is:$ (1) $\frac{7}{2 ^{20}}$ (2) $\frac{5}{2 ^{20}}$ (3) $\frac{4}{2 ^{20}}$ (4) $\frac{6}{2 ^{20}}$ 316. $If A, B are any two events of a random experiment and P (B) \neq = 1, then P (A ∖ B^{c}) =$ (1) $\frac{P ( A ) + P ( A \cap B )}{1 - P ( B )}$ (2) $\frac{P ( A ) - P ( A \cap B )}{1 - P ( B )}$ (3) $\frac{P ( A ) + P ( A \cap B )}{1 + P ( B )}$ (4) $\frac{P ( A )}{1 + P ( B )}$ 317. $If the probability distribution of a random variable X is given by X = x 0246810 P (X = x) 0 k 2 k 5 k^{2} 2 k^{2} 3 k then the mean of X is$ (1) $\frac{384}{121}$ (2) $\frac{60}{13}$ (3) $\frac{163}{25}$ (4) $\frac{326}{49}$ 318. $A number is selected at random from the set {1, 2, \dots .., 100} . Given that the number selected is divisible 2 , the probability that it is also divisible by 3 or 5 , is$ (1) $\frac{26}{50}$ (2) $\frac{23}{50}$ (3) $\frac{7}{50}$ (4) $\frac{13}{50}$ 319. $The probabilities of two persons to hit a target are \frac{1}{4} and \frac{1}{5} respectively. The probability that the target is being hit when both of them attempt independently is$ (1) $\frac{1}{2}$ (2) $\frac{3}{5}$ (3) $\frac{2}{5}$ (4) $\frac{7}{10}$ 320. $Two distinct numbers a and b are selected at random from 1, 2, 3, \dots, 50 . The probability, that their product ab is divisible by 3 , is$ (1) $\frac{272}{1225}$ (2) $\frac{561}{1225}$ (3) $\frac{664}{1225}$ (4) $\frac{8}{25}$ 321. $A multiple choice test consists of 5 questions, each question having 4 responses. There is only one correct response and the remaining 3 are incorrect responses. If a candidate attempts all the 5 questions then the probability that he answers at least 3 questions incorrectly is$ (1) $\frac{675}{1024}$ (2) $\frac{459}{512}$ (3) $\frac{81}{128}$ (4) $\frac{135}{512}$ 322. $In a workshop there are five machines and the probability of any one of them to be out of service on a day is \frac{1}{4} . If the probability that at most two machines will be out of service on the same day is (\frac{3}{4})^{3} k then k is equal to:$ (1) $\frac{17}{2}$ (2) $4$ (3) $\frac{17}{8}$ (4) $\frac{17}{7}$ 323. $From a pack of 52 cards, 3 cards are drawn at random. Then, the probability that one is ace, one is queen and one is jack is$ (1) $\frac{19}{5525}$ (2) $\frac{21}{5525}$ (3) $\frac{17}{5525}$ (4) $\frac{16}{5525}$ 324. $A, B, C are three horses participating in a face. The Probability of horse A to win the race is twice that of horse B and probability of horse B to win is twice that of horse C . Then the probabilities of horses A, B and C to win the race are respectively$ (1) $\frac{4}{7}, \frac{2}{7}, \frac{1}{7}$ (2) $\frac{1}{6}, \frac{2}{6}, \frac{5}{6}$ (3) $\frac{1}{4}, \frac{1}{2}, \frac{1}{7}$ (4) $\frac{4}{7}, \frac{3}{7}, \frac{1}{7}$ 325. $If the probability that an individual will suffer a bad reaction from an injection is 0.001 , then the probability that out of 2000 individuals, exactly 3 individuals suffer a bad reaction is$ (1) $\frac{4}{3 e ^{2}}$ (2) $\frac{2}{e ^{2}}$ (3) $\frac{2}{3 e ^{2}}$ (4) $\frac{4}{5 e ^{2}}$ 326. $If a binomial distribution B(n, p), the sum and product of the mean & variance are 5 and 6 respectively, then find 6(n+p-q) is equal to$ (1) $51$ (2) $52$ (3) $53$ (4) $50$ 327. $The p.m.f. of a random variable X is P (x) = (\frac{2 x}{n ( n + 1 )}, x = 1, 2, 3, \dots .. n 0, otherwise,) then E (X) is$ (1) $\frac{n + 1}{6}$ (2) $\frac{2 n + 1}{6}$ (3) $\frac{n + 1}{3}$ (4) $\frac{2 n + 1}{3}$ 328. $3 balls are drawn one after the other without replacement from an urn containing 4 red, 5 blue and 6 yellow balls. The probability of getting three different coloured balls is$ (1) $\frac{12}{9}$ (2) $\frac{24}{91}$ (3) $\frac{8}{225}$ (4) $\frac{8}{75}$ 329. $The probability that a non-leap year contains 53 Sunday is$ (1) $\frac{1}{7}$ (2) $\frac{1}{9}$ (3) $\frac{2}{7}$ (4) $\frac{1}{5}$ 330. $Let S be a set of 5 elements and P (S) denote the power set of S. Let E be an event of choosing an ordered pair (A, B) from the set P (S) \times P (S) such that A \cap B = \emptyset . If the probability of the event E is \frac{3 ^{p}}{2 ^{q}}, where p, q \in N, then p + q is equal to____$