Probability PYQs Set 14 (PDF)

Set 14 of 16 · Questions 391-420 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 14 (Q391-420)
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.
391. $Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is:$ (1) $\frac{275}{6 ^{5}}$ (2) $\frac{36}{5 ^{4}}$ (3) $\frac{181}{5 ^{5}}$ (4) $\frac{46}{6 ^{4}}$ 392. $Four fair dice D1, D2, D3 and D4 each having six faces numbered 1,2,3,4,5 and 6 are rolled simultaneously. The probability that D4 shows a number appearing on one of D1, D2 and D3 is$ (1) $\frac{91}{216}$ (2) $\frac{108}{216}$ (3) $\frac{125}{216}$ (4) $\frac{127}{216}$ 393. $If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is:$ (1) $\frac{1}{10}$ (2) $\frac{1}{5}$ (3) $\frac{3}{10}$ (4) $\frac{3}{20}$ 394. $Three numbers are chosen at random without replacement from {1, 2, 3, ..., 8}. The probability that their minimum is 3, given that their maximum is 6, is :$ (1) $\frac{3}{8}$ (2) $\frac{1}{5}$ (3) $\frac{1}{4}$ (4) $\frac{2}{5}$ 395. $Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is draw from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is:$ (1) $\frac{4}{9}$ (2) $\frac{5}{18}$ (3) $\frac{1}{6}$ (4) $\frac{3}{10}$ 396. $Let A and B be independent events such that P(A) = p,P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = \frac{5}{9}, is :$ (1) $\frac{1}{3}$ (2) $\frac{2}{9}$ (3) $\frac{4}{9}$ (4) $\frac{5}{12}$ 397. $A box contains 6 bottles of V_{1} drink, 3 bottles of V_{2} drink and 4 bottles of V_{3} drink. If three bottles are drawn at random, then the probability that the three are not of the same variety is$ (1) $\frac{632}{713}$ (2) $\frac{752}{833}$ (3) $\frac{833}{858}$ (4) $\frac{261}{286}$ 398. $A random variable ' X' has the following probability distribution : X P (X) 1 k - 1 2 3 k 3 k 4 3 k 5 3 k^{2} 6 k^{2} 7 k^{2} + k Then the value of k is$ (1) $\frac{1}{5}$ (2) $\frac{2}{7}$ (3) $- 2$ (4) $\frac{1}{10}$ 399. $A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is$ (1) $\frac{5}{7}$ (2) $\frac{2}{7}$ (3) $\frac{3}{7}$ (4) $47$ 400. $It is given that the discrete random variable X \sim B (n, p) and P (X = 2) = P (X = 3) . Then the mean of X is$ (1) $2 - p$ (2) $3 - p$ (3) $p - 2$ (4) $p - 3$ 401. $If A and B are two events such that P (A \cup B) = \frac{5}{6}, P (A ˉ) = \frac{1}{4} and P (B) = \frac{1}{3}, then A and B are$ (1) $mutually exclusive$ (2) $independent events$ (3) $exhaustive events$ (4) $exhaustive and independent events$ 402. $Let Ω be the sample space and A \subseteq Ω be an event. Given below are two statements: (S1) : If P (A) = 0, then A = ϕ (S2) : If P (A) = 1, then A = Ω Then$ (1) $Only (S1) is true$ (2) $Only (S2) is true$ (3) $both (S1) and (S2) are true$ (4) $Both (S1) and (S2) are false$ 403. $For two events A and B, a true statement among the following is$ (1) $P (A^{'} \cup B^{'}) = 1 - P (A) P (\frac{B}{A})$ (2) $P (A^{'} \cup B^{'}) = 1 - P (A \cup B)$ (3) $P (A^{'} \cup B^{'}) = P (A \cup B)$ (4) $P (A^{'} \cup B^{'}) = P (A^{'}) + P (B^{'})$ 404. $The probability distribution of a random variable X is given below: X: 4 k, \frac{30}{7} k, \frac{32}{7} k, \frac{34}{7} k, \frac{36}{7} k, \frac{38}{7} k, \frac{40}{7} k, 6 k P(X): \frac{2}{15}, \frac{1}{15}, \frac{2}{15}, \frac{1}{5}, \frac{1}{15}, \frac{2}{15}, \frac{1}{5}, \frac{1}{15} If E (X) = \frac{263}{15}, then P (X < 20) is equal to :$ (1) $\frac{11}{15}$ (2) $\frac{3}{5}$ (3) $\frac{14}{15}$ (4) $\frac{8}{15}$ 405. $Let A, B, C be three pairwise independent events of a random experiment. If P (B ˉ \cup C ˉ) = \frac{1}{2}, P (A) > 0, P (B) = b and P (C) = c, then P ((B ˉ \cap C ˉ) / A) =$ (1) $1 + b - c$ (2) $2 + b - c$ (3) $\frac{3}{2} - b - c$ (4) $2 - b - c$ 406. $For the random variable X with probability distribution given by the table X = x : 0, 1, 2, 3 P (X = x) : K, K + \frac{1}{7}, 2 K, \frac{2}{5} The mean of X is$ (1) $\frac{31}{35}$ (2) $\frac{57}{35}$ (3) $\frac{63}{35}$ (4) $\frac{67}{35}$ 407. $If X has a binomial distribution with parameters n = 6, p and P (X = 2) : P (X = 3) = 12 : 5, then p =$ (1) $\frac{5}{12}$ (2) $\frac{1}{2}$ (3) $\frac{16}{21}$ (4) $\frac{5}{21}$ 408. $In a city, 10 accidents take place in a span of 50 days. Assuming that the number of accidents follow the Poisson distribution the probability that three or more accidents occur in a day, is$ (1) $\sum_{k = 3}^{\infty} \frac{e ^{- λ} λ ^{k}}{k !}, λ = 0.2$ (2) $\sum_{k = 3}^{\infty} \frac{e ^{λ} λ ^{k}}{k !}, λ = 0.2$ (3) $1 - \sum_{k = 0}^{3} \frac{e ^{- λ} λ ^{k}}{k !}, λ = 0.2$ (4) $\sum_{k = 0}^{3} \frac{e ^{- λ} λ ^{k}}{k !}, λ = 0.2$ 409. $If A and B are two independent events such that P (A ˉ) = 0.75, P (A \cup B) = 0.65 and P (B) = x, then find the value of x .$ (1) $\frac{5}{14}$ (2) $\frac{9}{14}$ (3) $\frac{8}{15}$ (4) $\frac{7}{15}$ 410. $Let A and B be two events with P (A) = \frac{1}{7}, P (\frac{A}{B}) = \frac{2}{5} and P (B) = \frac{2}{7} . Then, the value P (\frac{B}{A}) is$ (1) $\frac{1}{5}$ (2) $\frac{5}{49}$ (3) $\frac{4}{5}$ (4) $\frac{3}{5}$ 411. $If A and B are two events such that P (A \cap B) = 0.1, and P (A ∣ B) and P (B ∣ A) are the roots of the equation 12 x^{2} - 7 x + 1 = 0, then the value of \frac{P ( A \cup B )}{P ( A \cap B )} is:$ (1) $\frac{4}{3}$ (2) $\frac{7}{4}$ (3) $\frac{5}{3}$ (4) $\frac{9}{4}$ 412. $If E and F are events such that P (F^{'}) = 0.7 and P (E \cap F) = 0.2, then P (E / F) is$ (1) $2/3$ (2) $1/3$ (3) $3/4$ (4) $1/4$ 413. $If S be the sample space of a random experiment ξ and P be a probability function defined on the power set P (S) of S, then which one of the following is not satisfied by P? (i) P (ϕ) = 0 (ii) If E^{c} is the complementary event of E, then P (E^{c}) = 1 - P (E) (iii) 0 \leq P (E) \leq 1, \forall E \subseteq S (iv) If E_{1} \subseteq E_{2} P (E_{2}) \leq P (E_{1})$ (1) $(iii)$ (2) $(iv)$ (3) $(ii)$ (4) $(i)$ 414. $In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35 %, 20 % and 10 % respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is:$ (1) $\frac{7}{45}$ (2) $\frac{8}{45}$ (3) $\frac{14}{45}$ (4) $\frac{28}{45}$ 415. $A shopkeeper buys a particular type of electric bulbs from three manufactures M_{1}, M_{2} and M_{3} . He buys 25 % of his requirement from M_{1}, 45 % from M_{2}, and 30 % from M_{3} . Based on past experience he found that 2 % of type M_{3} bulbs are defective, where as only 1 % of type M_{1} and type M_{2} are detective. If a bulb chosen by him at random is defective, then the probability that it was of type M_{3} is$ (1) $\frac{5}{13}$ (2) $\frac{6}{13}$ (3) $\frac{7}{13}$ (4) $\frac{8}{13}$ 416. $Cards are drawn one after the other without replacement from a well shuffled pack of cards until and ace card appears. If the probability that exactly 5 cards are drawn before the first ace card appears is \frac{4}{49} (\frac{p _{1} \cdot p _{2} \cdot p _{3}}{p _{4} \cdot p _{5} \cdot p _{6}}), (p_{i}) is prime, i = 1, 2, 3, 4, 5, 6) then (max (p_{i}) - min (p_{i})) =$ (1) $12$ (2) $18$ (3) $20$ (4) $22$ 417. $A company has two plants A and B to manufacture motorcycles. 60 % motorcycles are manufactured at plant A and the remaining are manufactured at plant B . 80 % of the motorcycles manufactured at plant A are rated of the standard quality, while 90 % of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126 p is$ (1) $54$ (2) $66$ (3) $64$ (4) $56$ 418. $If the probability that the random variable X takes values x is given by P (X = x) = k (x + 1) 3^{- x}, x = 0, 1, 2, 3, \dots, where k is a constant, then P (X \geq 2) is equal to$ (1) $\frac{7}{27}$ (2) $\frac{7}{18}$ (3) $\frac{11}{18}$ (4) $\frac{20}{27}$ 419. $The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is$ 420. $Of the three independent events E1, E2 and E3, the probability that only E1 occurs is α,only E2 occurs is β and only E3 occurs is γ. Let the probability p that none of events E1, E2 or E3 occurs satisfy the equations (α - 2β) p = αβ and (β - 3γ) p = 2βγ . All the given probabilities are assumed to lie in the interval (0, 1). Then \frac{P r o babi l i t y o f occ u r r e n ceo f E _{1}}{P r o babi l i t y o f occ u r r e n ceo f E _{3}} =$