Probability PYQs Set 13 (PDF)

Set 13 of 16 · Questions 361-390 of 458

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 13 (Q361-390)
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361. $An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is$ (1) $\frac{2}{7}$ (2) $\frac{1}{21}$ (3) $\frac{2}{23}$ (4) $\frac{1}{3}$ 362. $Five numbers x_{1}, x_{2}, x_{3}, x_{4}, x_{5} are randomly selected from the numbers 1, 2, 3,......, 18 and are arranged in the increasing order (x_{1} < x_{2} < x_{3} < x_{4} < x_{5}). The probability that x_{2} = 7 and x_{4} = 11 is :$ (1) $\frac{1}{136}$ (2) $\frac{1}{72}$ (3) $\frac{1}{68}$ (4) $\frac{1}{34}$ 363. $Let E_{1}, E_{2}, E_{3} be three mutually exclusive events such that P (E_{1}) = \frac{2 + 3 p}{6}, P (E_{2}) = \frac{2 - p}{8} and P(E_{3}) = \frac{1 - p}{2} . If the maximum and minimum values of p are p_{1} and p_{2}, then (p_{1} + p_{2}) is equal to :$ (1) $\frac{2}{3}$ (2) $\frac{5}{3}$ (3) $\frac{5}{4}$ (4) $1$ 364. $The random variable X follows binomial distribution B(n, p), for which the difference of the mean and the variance is 1. If 2 P(X = 2) = 3 P(X = 1), then n^{2} P(X > l) is equal to$ (1) $15$ (2) $11$ (3) $12$ (4) $16$ 365. $Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is :$ (1) $\frac{2}{3}$ (2) $\frac{11}{16}$ (3) $\frac{23}{32}$ (4) $\frac{13}{16}$ 366. $A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing atleast 'n' correct answers is less than \frac{1}{2}, is$ (1) $5$ (2) $6$ (3) $3$ (4) $4$ 367. $Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2x2 matrices. The probability that such formed matrices have all different entries and are non-singular, is :$ (1) $\frac{46}{162}$ (2) $\frac{23}{81}$ (3) $\frac{22}{81}$ (4) $\frac{43}{162}$ 368. $The probability that a randomly selected 2 -digit number belongs to the set {n \inN : (2^{n} - 2) is a multiple of 3} is equal to$ (1) $\frac{1}{6}$ (2) $\frac{2}{3}$ (3) $\frac{1}{2}$ (4) $\frac{1}{3}$ 369. $If a point A(x, y) lies in the region bounded by the y-axis, straight lines 2y + x = 6 and 5x - 6y = 30, then the probability that y < 1 is :$ (1) $\frac{1}{6}$ (2) $\frac{5}{6}$ (3) $\frac{2}{3}$ (4) $\frac{6}{7}$ 370. $A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is$ (1) $\frac{38}{16}$ (2) $\frac{15}{16}$ (3) $\frac{21}{16}$ (4) $\frac{81}{64}$ 371. $If a random variable X follows the Binomial distribution B (33, p) such that 3P(X = 0) = P(X = 1), then the value of \frac{P _{(} X = 1 5 _{)}}{P _{(} X = 1 8 _{)}} - \frac{P _{(} X = 1 6 _{)}}{P _{(} X = 1 7 _{)}} is equal to$ (1) $1320$ (2) $1088$ (3) $\frac{120}{1331}$ (4) $\frac{1088}{1089}$ 372. $The probability, that in a randomly selected 3-digit number at least two digits are odd, is$ (1) $\frac{19}{36}$ (2) $\frac{15}{36}$ (3) $\frac{13}{36}$ (4) $\frac{23}{36}$ 373. $Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If P(X > n - 3) = \frac{k}{2 ^{n}}, then k is equal to$ (1) $528$ (2) $529$ (3) $629$ (4) $630$ 374. $Let X be a binomially distributed random variable with mean 4 and variance \frac{4}{3} . Then 54 P(X \leq 2) is equal to$ (1) $\frac{73}{27}$ (2) $\frac{146}{27}$ (3) $\frac{146}{81}$ (4) $\frac{126}{81}$ 375. $The mean and variance of a binomial distribution are α and \frac{α}{3} respectively. If P (X = l) = \frac{4}{243}, then P(X = 4 or 5) is equal to :$ (1) $\frac{5}{9}$ (2) $\frac{64}{81}$ (3) $\frac{16}{27}$ (4) $\frac{145}{243}$ 376. $A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is :$ (1) $\frac{17}{3 ^{5}}$ (2) $\frac{13}{3 ^{5}}$ (3) $\frac{11}{3 ^{5}}$ (4) $\frac{10}{3 ^{5}}$ 377. $Two dice A and B are rolled. Let the numbers obtained on A and B be α and β respectively. If the variance of α - β is \frac{p}{q}, where p and g are co-prime, then the sum of the positive divisors of p is equal to$ (1) $72$ (2) $36$ (3) $48$ (4) $31$ 378. $Bag A contains 4 white and 2 black balls, bag B contains 3 white and 3 black balls and bag C contains 2 white and 4 black balls. If a bag is chosen at random and a ball is chosen at random from it, then the probability that the ball drawn is black is$ (1) $\frac{1}{3}$ (2) $\frac{3}{4}$ (3) $\frac{1}{2} 1$ (4) $\frac{2}{3}$ 379. $A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is$ (1) $\frac{1}{4}$ (2) $\frac{11}{50}$ (3) $\frac{1}{5}$ (4) $\frac{9}{50}$ 380. $A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is$ (1) $\frac{3}{4}$ (2) $\frac{3}{10}$ (3) $\frac{2}{5}$ (4) $\frac{1}{5}$ 381. $If C and D are two events such that C \subset D and P(d) \neq 0, then the correct statement among the following is :$ (1) $P(C\|D) = P(c)$ (2) $P(C\|D) \geq P(c)$ (3) $P(C\|D) < P(c)$ (4) $P(C\|D) = \frac{P ( D )}{P ( C )}$ 382. $3 out of 6 vertices of a regular hexagon are chosen at a time at random. The probability that the triangle formed with these three vertices is an equilateral triangle, is$ (1) $\frac{1}{2}$ (2) $\frac{1}{5}$ (3) $\frac{1}{10}$ (4) $\frac{1}{20}$ 383. $In a binomial distribution B (n, p = \frac{1}{4}), if the probability of at least one success is greater than or equal to \frac{9}{10}, then n is greater than :$ (1) $\frac{1}{l o g _{10} 4 + l o g _{10} 3}$ (2) $\frac{9}{l o g _{10} 4 - l o g _{10} 3}$ (3) $\frac{4}{l o g _{10} 4 - l o g _{10} 3}$ (4) $\frac{1}{l o g _{10} 4 - l o g _{10} 3}$ 384. $In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective then the probability that it is manufactured by the machine C is$ (1) $\frac{5}{14}$ (2) $\frac{9}{28}$ (3) $\frac{3}{7}$ (4) $\frac{2}{7}$ 385. $Let S = {E,....} be a sample space of random experiment such that P()= for every n = 1, 2....S. Then the number of elements in the set is _______$ 386. $Let A be the event that the absolute difference between two randomly choosen real numbers in the sample space (0, 60) is less than or equal to a. If P(A) = \frac{11}{36}, then a is equal to _________$ 387. $A fair n (n > 1) faces die is rolled repeatedly until a number less than n appears. If the mean of the number of tosses required is \frac{n}{9}, then n is equal to$ 388. $Let X be a random variable with distribution. If the mean of X is 2.3 and variance of X is σ^{2},then 100 σ^{2} is equal to :$ 389. $The probability distribution of random variable X is given by: Let p = P(1 < X < 4 ∣ X < 3). If 5p = λK, then λ equal to .$ 390. $Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4 . If the probability that a x^{2} + bx + c = 0 has all real roots is \frac{m}{n}, gcd (m, n) = 1, then m + n is equal to__________$