Probability PYQs Set 8 (PDF)

Set 8 of 16 · Questions 211-240 of 458

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 8 (Q211-240)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
211. $Let A denote the event that a 6 -digit integer formed by 0,1,2,3,4,5,6 without repetitions,be divisible by 3.Then probability of eventA is equal to:$ (1) $\frac{9}{56}$ (2) $\frac{4}{9}$ (3) $\frac{3}{7}$ (4) $\frac{11}{27}$ 212. $Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable x to be the number of rotten apples in a draw of two apples, the variance of x is$ (1) $\frac{37}{153}$ (2) $\frac{57}{153}$ (3) $\frac{47}{153}$ (4) $\frac{40}{153}$ 213. $A random variable X has the range 0, 1, 2, \dots . If P (X = r) = k (1 + r) 3^{- r}, for r = 0, 1, 2, \dots . where k > 0 is a real number, then P (X = 0) + P (X = 1) + P (X = 2) =$ (1) $\frac{4}{9}$ (2) $\frac{8}{9}$ (3) $\frac{2}{3}$ (4) $\frac{1}{3}$ 214. $In a binomial distribution, mean is 18 and variance is 12 then p =$ (1) $2/3$ (2) $1/3$ (3) $¾$ (4) $1/2$ 215. $If the probability function of a random variable X is defined by P (X = k) = a (\frac{k + 1}{2 ^{k}}) for k = 0, 1, 2, 3, 4, 5, then the probability that X takes a prime value is$ (1) $\frac{13}{20}$ (2) $\frac{23}{60}$ (3) $\frac{11}{20}$ (4) $\frac{19}{60}$ 216. $Given below is the distribution of a random variable X X = x 1234 P (X = x) λ 2 λ 3 λ 4 λ If α = P (X < 3) and β = P (X > 2), then α : β =$ (1) $2 : 5$ (2) $3 : 4$ (3) $4 : 5$ (4) $3 : 7$ 217. $Three students X, Y and Z appear for an examination. The probability of X passing the examination is \frac{1}{5}, the probability of Y passing the examination is \frac{1}{4} and the probability of Z failing the examination is \frac{2}{3} . The probability that atleast two of them pass the exam is$ (1) $\frac{1}{6}$ (2) $\frac{2}{5}$ (3) $\frac{3}{4}$ (4) $\frac{3}{5}$ 218. $In a random experiment of throwing 5 coins, the number of heads is defined as a random variable. The mean of the random variable is$ (1) $\frac{2}{3}$ (2) $\frac{3}{2}$ (3) $\frac{7}{9}$ (4) $\frac{5}{2}$ 219. $In a committee of 25 members, each member is proficient either in Mathematics or in Statistics or in both. If 19 of them are proficient in Mathematics and 16 of them are proficient in Statistics, then the probability that a person selected at random from the committee is proficient in both is$ (1) $\frac{1}{5}$ (2) $\frac{3}{5}$ (3) $\frac{2}{5}$ (4) $\frac{1}{2}$ 220. $An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :$ (1) $\frac{5}{256}$ (2) $\frac{5}{715}$ (3) $\frac{3}{715}$ (4) $\frac{3}{256}$ 221. $Bag B_{1} contains 4 white and 2 black balls. Bag B_{2} contains 3 white and 4 black balls. A bag is chosen at random and a ball is drawn from it at random, then the probability that the ball drawn is white, is$ (1) $\frac{1}{42}$ (2) $\frac{42}{32}$ (3) $\frac{33}{42}$ (4) $\frac{23}{42}$ 222. $Two letters are chosen from the letters of the word 'EQUATIONS'. The probability that one is vowel and the other is consonant is$ (1) $\frac{8}{9}$ (2) $\frac{3}{9}$ (3) $\frac{4}{9}$ (4) $\frac{5}{9}$ 223. $On an average nine out of 10 ships that have departed at A reach B safely. The probability that out of five ships that have departed at A atleast four will reach B safely is$ (1) $14 (0.9)^{5}$ (2) $1.4 (0.9)^{5}$ (3) $0.14 (0.9)^{4}$ (4) $1.4 (0.9)^{4}$ 224. $Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of 4. If p is the conditional probability that number 4 has appeared atleast once, then 3 p + 2 =$ (1) $\frac{25}{12}$ (2) $\frac{1}{6}$ (3) $\frac{7}{3}$ (4) $\frac{5}{2}$ 225. $Two dice are thrown independently. Let A be the event that the number appeared on the 1^{s t} die is less than the number appeared on the 2^{n d} die. B be the even and that on the second die is odd, and C be the event that the number appeared on the 1^{s t} die is odd and that on the 2^{n d} is even. Then$ (1) $The number of favorable cases of the event (A \cup B) \cap C is 6$ (2) $A and B are mutually excusive$ (3) $The number of favorable cases of the events A, B and C are 15, 6 and 6 respectively$ (4) $B and C are independent$ 226. $5 persons entered a lift cabin in the cellar of a 7 -floor building apart from cellar. If each of them independently and with equal probability can leave the cabin at any floor out of the 7 floors beginning with the first, then the probability of all the 5 persons leaving the cabin at different floors is$ (1) $\frac{360}{2401}$ (2) $\frac{5}{54}$ (3) $\frac{51}{71}$ (4) $\frac{5}{18}$ 227. $If A, B and C are three independent events such that P (A) = P (B) = P (C) = p, then P (at least two of A, B, C occur) =$ (1) $p^{3} - 3 p$ (2) $3 p^{2} - 2 p^{3}$ (3) $3 p - 2 p^{2}$ (4) $3 p^{2}$ 228. $If a random variable x has the probability distribution X P (x) 001 2k 2 K 3 3 k 4 2 k^{2} 5 2 k 6 k^{2} + k 7 7 k^{2} then P (3 < x \leq 6) is equal to$ (1) $0.64$ (2) $0.22$ (3) $0.33$ (4) $0.34$ 229. $Two events A and B will be independent if$ (1) $P (A^{’} \cap B^{’}) = (1 - P (A)) (1 - P (B))$ (2) $A and B are mutually exclusive$ (3) $P (A) + P (B) = 1$ (4) $P (A) = P (B)$ 230. $Let A = (a_{ij}) be a square matrix of order 2 with entries either 0 or 1. Let E be the event that A is an invertible matrix. Then the probability P (E) is:$ (1) $\frac{3}{16}$ (2) $\frac{5}{8}$ (3) $\frac{3}{8}$ (4) $\frac{1}{8}$ 231. $In a non-leap year, the probability of getting 53 Sunday or 53 Tuesday or 53 Thursday is$ (1) $\frac{1}{7}$ (2) $\frac{2}{7}$ (3) $\frac{3}{7}$ (4) $\frac{4}{7}$ 232. $For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problems is \frac{4}{5}, then the probability, that he is unable to solve less than two problems is$ (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) $\frac{164}{25} (\frac{1}{5})^{48}$ 233. $Two cards are drawn simultaneously from a well shuffled deck of 52 cards. Find the probability distribution of number of aces.$ (1) $X 012 P(X) 62/221 74/221 32/221$ (2) $X 012 P(X) 188/221 32/221 1/221$ (3) $X 012 P(X) 4/221 16/221 3/221$ (4) $X 012 P(X) 54/221 72/221 10/221$ 234. $Which of the following is not a property of a Binomial distribution?$ (1) $Random experiment consists of a sequence of n identical trials$ (2) $Each outcome can be referred to as a success or a failure$ (3) $The probabilities of the two outcomes can change from one trial to the next$ (4) $The trial are independent$ 235. $When a pair of six faced fair dice are thrown, the probability that the sum of the numbers on the two dice is greater than 7 , is$ (1) $\frac{1}{3}$ (2) $\frac{5}{12}$ (3) $\frac{1}{2}$ (4) $\frac{1}{4}$ 236. $The probability distribution of X is X 0123 P (X) 0.3 k 2 k 2 k The value of k is$ (1) $0.7$ (2) $0.3$ (3) $1$ (4) $0.14$ 237. $If X is a binomial variate with mean 6 and variance 2 , then the value of P (5 \leq X \leq 7) is$ (1) $\frac{4762}{6561}$ (2) $\frac{4672}{6561}$ (3) $\frac{5264}{6561}$ (4) $\frac{5462}{6651}$ 238. $Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is \frac{m}{n}, where gcd (m, n) = 1, then m + n is equal to:$ (1) $4$ (2) $14$ (3) $13$ (4) $11$ 239. $Three distinct numbers are selected randomly from the set 1, 2, 3, \dots, 40 . If the probability, that the selected numbers are in an increasing G.P., is \frac{m}{n}, gcd (m, n) = 1, then m + n is equal to _____.$ 240. $Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y respectively denote the number of blue and yellow balls. If X and Y are the means of X and Y respectively, then 7 X + 4 Y is equal to________$