Probability PYQs Set 1 (PDF)

Set 1 of 16 · Questions 1-30 of 458

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Probability · Set 1 (Q1-30)
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1. $Let E^{C} denote the complement of an event E . Let E_{1}, E_{2} and E_{3} be any pairwise independent events with P (E_{1}) > 0 and P (E_{1} \cap E_{2} \cap E_{3}) = 0 Then P (E_{2}^{C} \cap E_{3}^{C} / E_{1}) is equal to:$ (1) $P (E_{3}^{C}) - P (E_{2}^{C})$ (2) $P (E_{2}^{C}) + P (E_{3})$ (3) $P (E_{3}) - P (E_{2}^{C})$ (4) $P (E_{3}^{C}) - P (E_{2})$ 2. $Probability of guessing correctly atleast 7 out of 10 answers in a "True" or "False" test is$ (1) $\frac{11}{64}$ (2) $\frac{11}{32}$ (3) $\frac{11}{16}$ (4) $\frac{27}{32}$ 3. $Given three indentical bags each containing 10 Balls, whose colours are as follows: Red Blue Green Bag I 325 Bag II 433 Bag III 514 A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is q, then the value of (\frac{1}{p} + \frac{1}{q}) is:$ (1) $6$ (2) $9$ (3) $7$ (4) $8$ 4. $A target is to be destroyed in a bombing exercise and there is a 75 % chance that a bomb will hit the target. Assuming that two direct hits are required to destroy the target completely, the minimum number of bombs to be dropped in order that the probability of destroying the target is not less than 99 %, is$ (1) $4$ (2) $5$ (3) $6$ (4) $7$ 5. $If the mean and variance of a binomial distribution are 4 and \frac{4}{3} respectively, then P (X = 2) =$ (1) $\frac{20}{243}$ (2) $\frac{40}{243}$ (3) $\frac{28}{729}$ (4) $\frac{8}{27}$ 6. $A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is:$ (1) $\frac{2}{5}$ (2) $\frac{2}{7}$ (3) $\frac{1}{7}$ (4) $\frac{1}{5}$ 7. $A and B each select one number at random from the distinct numbers 1, 2, 3, \dots, n and the probability that the number selected by A is less than the number selected by B is \frac{1009}{2019} . Now, the probability that the number selected by B is the number immediately next to the number selected by A is$ (1) $\frac{2018}{2019}$ (2) $\frac{2018}{( 2019 ) ^{2}}$ (3) $\frac{2000}{( 2019 )}$ (4) $\frac{2000}{( 2019 ) ^{2}}$ 8. $The probability of selecting integers a \in [-5, 30] such that x^{2} + 2(a + 4)x - 5a + 64 > 0, for all x \in R, is:$ (1) $\frac{7}{36}$ (2) $\frac{2}{9}$ (3) $\frac{1}{6}$ (4) $\frac{1}{4}$ 9. $One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is$ (1) $\frac{2}{3}$ (2) $\frac{1}{2}$ (3) $\frac{4}{9}$ (4) $\frac{3}{5}$ 10. $When a missile is fired from a ship, the probability that it is intercepted is \frac{1}{3} and the probability that the missile hits the target, given that it is not intercepted, is \frac{3}{4} . If three missiles are fired independently from the ship, then the probability that all three hit the target, is:$ (1) $\frac{1}{8}$ (2) $\frac{1}{27}$ (3) $\frac{3}{4}$ (4) $\frac{3}{8}$ 11. $If A, B, C are three mutually exclusive and exhaustive events of an experiment such that P (A) = 2 P (B) = 3 P (C), then P (B) is equal to$ (1) $\frac{2}{11}$ (2) $\frac{3}{11}$ (3) $\frac{4}{11}$ (4) $\frac{1}{11}$ 12. $Given that A and B are two events such that P (B) = \frac{3}{5}, P (\frac{A}{B}) = \frac{1}{2} and P (A \cup B) = \frac{4}{5}, then P (A) =$ (1) $\frac{3}{10}$ (2) $\frac{1}{2}$ (3) $\frac{1}{5}$ (4) $\frac{3}{5}$ 13. $Let A and B be two independent events such that P (A) = \frac{1}{3} and P (B) = \frac{1}{6} Then, which of the following is TRUE?$ (1) $P (\frac{A}{B}) = \frac{1}{6}$ (2) $P (\frac{A}{B ^{’}}) = \frac{1}{3}$ (3) $P (\frac{A}{B ^{’}}) = \frac{2}{3}$ (4) $P (\frac{A}{B}) = \frac{5}{6}$ 14. $Four persons can hit a target correctly with probabilities \frac{1}{2}, \frac{1}{3}, \frac{1}{4} and \frac{1}{8} respectively. If all hit at the targetindependently, then the probability that the target would be hit, is:$ (1) $\frac{25}{192}$ (2) $\frac{7}{32}$ (3) $\frac{1}{192}$ (4) $\frac{25}{32}$ 15. $A person P speaks truth in 75 % cases and another person R in 80 % cases. Then the probability that they likely to contradict each other in narrating the same event is$ (1) $\frac{7}{20}$ (2) $\frac{7}{10}$ (3) $0.2$ (4) $0.3$ 16. $If X_{1}, X_{2}, \dots X_{n} are n independent events such that P (X_{r}) = \frac{1}{r + 1}, r = 1, 2, \dots, n, then the probability that none of the n events occur is$ (1) $\frac{1}{n}$ (2) $\frac{1}{n + 1}$ (3) $\frac{n}{n + 1}$ (4) $\frac{n + 1}{n + 2}$ 17. $Three critics review a book. For the three critics the odds in favour of the book are 2 : 5, 3 : 4 and 4 : 3 respectively. The probability that the majority is in favour of the book, is given by$ (1) $\frac{183}{343}$ (2) $\frac{160}{343}$ (3) $\frac{209}{343}$ (4) $\frac{134}{343}$ 18. $If a cubical die is thrown, then the mean and variance of the random variable X, giving the number on the face that shows up, are respectively$ (1) $\frac{2}{7}, \frac{12}{35}$ (2) $\frac{7}{2}, \frac{12}{35}$ (3) $\frac{1}{7}, \frac{1}{12}$ (4) $\frac{7}{2}, \frac{35}{12}$ 19. $If getting a head on a coin when it is tossed is considered as success, then the probability of having more number of failures when ten fair coins are tossed simultaneously, is$ (1) $\frac{105}{2 ^{8}}$ (2) $\frac{73}{2 ^{7}}$ (3) $\frac{193}{2 ^{9}}$ (4) $\frac{638}{2 ^{10}}$ 20. $Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn in white, is :$ (1) $\frac{1}{4}$ (2) $\frac{1}{9}$ (3) $\frac{1}{3}$ (4) $\frac{3}{10}$ 21. $A bag contains 4 red and 6 balck balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is$ (1) $\frac{1}{5}$ (2) $\frac{3}{4}$ (3) $\frac{3}{10}$ (4) $\frac{2}{5}$ 22. $Two balls are drawn at random from a bag containing 5 black balls and 3 white balls. If the random variable X denotes the number of white balls drawn, then the mean of X is.$ (1) $\frac{1}{2}$ (2) $\frac{5}{8}$ (3) $\frac{3}{4}$ (4) $\frac{3}{8}$ 23. $Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is:$ (1) $\frac{1}{66}$ (2) $\frac{1}{11}$ (3) $\frac{1}{9}$ (4) $\frac{2}{11}$ 24. $A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of 4 . Then the conditional probability that the score 4 has appeared atleast once is:$ (1) $\frac{1}{3}$ (2) $\frac{1}{4}$ (3) $\frac{1}{8}$ (4) $\frac{1}{9}$ 25. $In an experiment a person gets success α times out of β trails. If the experiment consists of n trials, then the probability that he fails at least (n - 1) times is$ (1) $\frac{α ^{n - 1}}{β ^{n}} (nβ - nα + α)$ (2) $\frac{( β - α ) ^{n - 1}}{β ^{n}} (nα + β - α)$ (3) $\frac{α ^{n}}{β ^{n}} (nα + β)$ (4) $(\frac{β - α}{β})^{n} (nβ + nα + 1) - 0$ 26. $The variance of a Poisson variate X is 2 Then P (X \geq 3) =$ (1) $\frac{e ^{2} - 7}{e ^{2}}$ (2) $\frac{e ^{2} - 3}{e ^{2}}$ (3) $\frac{e ^{2} - 5}{e ^{2}}$ (4) $1 - \frac{4}{e ^{2}}$ 27. $A bag contains four balls. Two balls are drawn randomly and found them to be white. The probability that all the balls in the bag are white is$ (1) $\frac{1}{2}$ (2) $\frac{3}{5}$ (3) $\frac{1}{4}$ (4) $\frac{2}{3}$ 28. $Let X \sim B (n, p), if E (X) = 5, Var (X) = 2.5 then P (X < 1) =$ (1) $(\frac{1}{2})^{11}$ (2) $(\frac{1}{2})^{10}$ (3) $(\frac{1}{2})^{6}$ (4) $(\frac{1}{2})^{9}$ 29. $A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability, that A wins if A makes the first throw, is$ (1) $\frac{8}{17}$ (2) $\frac{9}{19}$ (3) $\frac{9}{17}$ (4) $\frac{8}{19}$ 30. $Four fair dice are thrown independently 27 times. Then the expected number of times, at least twodice show up a three or a five, is$