JEE Mathematics Probability MCQs Set E

Type – 1

Choose the most appropriate option (a, b, c or d).

Question. Five boys three girls are seated at random in a row. The probability that no boy sits between girls is
(a) \( \frac{1}{56} \)
(b) \( \frac{1}{8} \)
(c) \( \frac{3}{28} \)
(d) None of the options
Answer: (c) \( \frac{3}{28} \)

Question. In a convex hexagon two diagonals are drawn at random. The probability that the diagonals intersect at an interior point of the hexagon is
(a) \( \frac{5}{12} \)
(b) \( \frac{7}{12} \)
(c) \( \frac{2}{5} \)
(d) None of the options
Answer: (a) \( \frac{5}{12} \)

Question. 4 five–rupee coins, 3 two–rupee coins and 2 one–rupee coins are stacked together in a column at random. The probability that the coins of the same denomination are consecutive is
(a) \( \frac{13}{9!} \)
(b) \( \frac{1}{210} \)
(c) \( \frac{1}{35} \)
(d) None of the options
Answer: (b) \( \frac{1}{210} \)

Question. Two cards are drawn at random from a pack of 52 cards. The probability that they contain at least a spade and an ace is
(a) \( \frac{1}{34} \)
(b) \( \frac{8}{221} \)
(c) \( \frac{1}{26} \)
(d) \( \frac{2}{51} \)
Answer: (c) \( \frac{1}{26} \)

Question. A five–digit number is written down at random. The probability that the number is divisible by 5 and no two consecutive digits are identical, is
(a) \( \frac{1}{5} \)
(b) \( \frac{1}{5} \cdot \left( \frac{9}{10} \right)^3 \)
(c) \( \left( \frac{3}{5} \right)^4 \)
(d) None of the options
Answer: (c) \( \left( \frac{3}{5} \right)^4 \)

Question. If the letters of the words ATTEMPT are written down at random, the chance that all Ts are consecutive is
(a) \( \frac{1}{42} \)
(b) \( \frac{6}{7} \)
(c) \( \frac{1}{7} \)
(d) None of the options
Answer: (c) \( \frac{1}{7} \)

Question. In a single cast with two dice the odds against drawing 7 is
(a) \( \frac{1}{6} \)
(b) \( \frac{1}{12} \)
(c) 5 : 1
(d) 1 : 5
Answer: (c) 5 : 1

Question. 7 white balls and 3 black balls are placed in a row at random. The probability that no two black balls are adjacent is
(a) \( \frac{1}{2} \)
(b) \( \frac{7}{15} \)
(c) \( \frac{2}{15} \)
(d) \( \frac{1}{3} \)
Answer: (b) \( \frac{7}{15} \)

Question. 10 apples are distributed at random among 6 person. The probability that at least one of them will receive none is
(a) \( \frac{6}{143} \)
(b) \( \frac{{}^{14}C_4}{{}^{15}C_5} \)
(c) \( \frac{137}{143} \)
(d) None of the options
Answer: (c) \( \frac{137}{143} \)

Question. 4 gentlemen and 4 ladies take seats at random round a table. The probability that they are sitting alternately is
(a) \( \frac{4}{35} \)
(b) \( \frac{1}{70} \)
(c) \( \frac{2}{35} \)
(d) \( \frac{1}{35} \)
Answer: (d) \( \frac{1}{35} \)

Question. Let \( x = 33^n \). The index \( n \) is given a positive integral value at random. The probability that the value of \( x \) will have 3 in the units place is
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{3} \)
(d) None of the options
Answer: (a) \( \frac{1}{4} \)

Question. Three dice are thrown simultaneously. The probability of getting a sum of 15 is
(a) \( \frac{1}{72} \)
(b) \( \frac{5}{36} \)
(c) \( \frac{5}{72} \)
(d) None of the options
Answer: (d) None of the options

Question. Three dice are thrown. The probability of getting a sum which is a perfect square is
(a) \( \frac{2}{5} \)
(b) \( \frac{9}{20} \)
(c) \( \frac{1}{4} \)
(d) None of the options
Answer: (d) None of the options

Question. The probability of getting a sum of 12 in four throws of an ordinary dice is
(a) \( \frac{1}{6} \left( \frac{5}{6} \right)^3 \)
(b) \( \left( \frac{5}{6} \right)^4 \)
(c) \( \frac{1}{36} \left( \frac{5}{6} \right)^2 \)
(d) None of the options
Answer: (a) \( \frac{1}{6} \left( \frac{5}{6} \right)^3 \)

Question. Three different numbers are selected at random from the set A = {1, 2, 3, ..... ,10}. The probability that the product of two of the numbers is equal to the third is
(a) \( \frac{3}{4} \)
(b) \( \frac{1}{40} \)
(c) \( \frac{1}{8} \)
(d) None of the options
Answer: (b) \( \frac{1}{40} \)

Question. There are 7 seats in a row. Three persons take seats at random. The probability that the middle seat is always occupied and no two persons are consecutive is
(a) \( \frac{9}{70} \)
(b) \( \frac{9}{35} \)
(c) \( \frac{4}{35} \)
(d) None of the options
Answer: (c) \( \frac{4}{35} \)

Question. A second-order determinant is written down at random using the numbers 1, -1 as elements. The probability that the value of the determinant is non-zero is
(a) \( \frac{1}{2} \)
(b) \( \frac{3}{8} \)
(c) \( \frac{5}{8} \)
(d) \( \frac{1}{3} \)
Answer: (a) \( \frac{1}{2} \)

Question. \( x_1, x_2, x_3, ... x_{50} \) are fifty real numbers such that \( x_r < x_{r+1} \) for \( r = 1, 2, 3, ..., 49 \). Five numbers out of these are picked up at random. Then probability that the five numbers have \( x_{20} \) as the middle number is
(a) \( \frac{{}^{20}C_2 \times {}^{30}C_2}{{}^{50}C_5} \)
(b) \( \frac{{}^{30}C_2 \times {}^{19}C_2}{{}^{50}C_5} \)
(c) \( \frac{{}^{19}C_2 \times {}^{31}C_3}{{}^{50}C_5} \)
(d) None of the options
Answer: (b) \( \frac{{}^{30}C_2 \times {}^{19}C_2}{{}^{50}C_5} \)

Question. Numbers 1, 2, 3, ... , 100 are written down on each of the cards A, B and C. One number is selected at random from each of the cards. Then probability that the numbers so selected can be the measures (in cm) of three sides of right-angled triangles no two of which are similar, is
(a) \( \frac{4}{100^3} \)
(b) \( \frac{3}{50^3} \)
(c) \( \frac{3!}{100^3} \)
(d) None of the options
Answer: (d) None of the options

Question. Three numbers are chosen at random without replacement from the set A = {x | 1 ≤ x ≤ 10, x ∈ N). The probability that the minimum of the chosen numbers is 3 and maximum is 7, is
(a) \( \frac{1}{12} \)
(b) \( \frac{1}{15} \)
(c) \( \frac{1}{40} \)
(d) None of the options
Answer: (c) \( \frac{1}{40} \)

Question. Three natural numbers are taken at random from the set A = {x | 1 ≤ x ≤ 100, x ∈ N}. The probability that the AM of the numbers taken is 25 is
(a) \( \frac{{}^{77}C_2}{{}^{100}C_3} \)
(b) \( \frac{{}^{25}C_2}{{}^{100}C_3} \)
(c) \( \frac{{}^{74}C_{72}}{{}^{100}C_{97}} \)
(d) None of the options
Answer: (c) \( \frac{{}^{74}C_{72}}{{}^{100}C_{97}} \)

Question. Let S be the universal set and \( n(X) = k \). The probability of selecting two subsets A and B of the set X such that \( B = \bar{A} \) is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{2^k - 1} \)
(c) \( \frac{1}{2^k} \)
(d) \( \frac{1}{3^k} \)
Answer: (b) \( \frac{1}{2^k - 1} \)

Question. From a group of 10 persons consisting of 5 lawyers, 3 doctors and 2 engineers, four persons are selected at random. The probability that the selection contains at least one of each category is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{2}{3} \)
(d) None of the options
Answer: (a) \( \frac{1}{2} \)

Question. 10 different books and 2 different pens are given to 3 boys so that each gets equal number of things. The probability that the same boy does not receive both the pens is
(a) \( \frac{5}{11} \)
(b) \( \frac{7}{11} \)
(c) \( \frac{2}{3} \)
(d) \( \frac{6}{11} \)
Answer: (d) \( \frac{6}{11} \)

Question. Two distinct numbers are selected at random from the first twelve natural numbers. The probability that the sum will be divisible by 3 is
(a) \( \frac{1}{3} \)
(b) \( \frac{23}{66} \)
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (a) \( \frac{1}{3} \)

Question. The probability of a number \( n \) showing in a throw of a dice marked 1 to 6 is proportional to \( n \). Then the probability of the number 3 showing in a throw is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{6} \)
(c) \( \frac{1}{7} \)
(d) \( \frac{1}{21} \)
Answer: (c) \( \frac{1}{7} \)

Question. The probability that out of 10 persons, all born in April, at least two have the same birthday is
(a) \( \frac{{}^{30}C_{10}}{(30)^{10}} \)
(b) \( 1 - \frac{{}^{30}C_{10}}{30!} \)
(c) \( 1 - \frac{{}^{30}P_{10}}{30^{10}} \)
(d) None of the options
Answer: (c) \( 1 - \frac{{}^{30}P_{10}}{30^{10}} \)

Question. If one ball is drawn at random from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls then the probability that 2 white and 1 black balls will be drawn is
(a) \( \frac{13}{32} \)
(b) \( \frac{1}{4} \)
(c) \( \frac{1}{32} \)
(d) \( \frac{3}{16} \)
Answer: (a) \( \frac{13}{32} \)

Question. A draws two cards at random from a pack of 52 cards. After returning them to the pack and shuffling it, B draws two cards at random. The probability that their draws contain exactly one common card is
(a) \( \frac{25}{546} \)
(b) \( \frac{50}{663} \)
(c) \( \frac{25}{663} \)
(d) None of the options
Answer: (b) \( \frac{50}{663} \)

Question. A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is
(a) \( \frac{44}{85 \times 49} \)
(b) \( \frac{11}{85 \times 49} \)
(c) \( \frac{13 \times 24}{17 \times 25 \times 49} \)
(d) None of the options
Answer: (a) \( \frac{44}{85 \times 49} \)

Question. Three numbers are chosen at random without replacement from 1, 2, 3, ..., 10. The probability that the minimum of the chosen numbers is 4 or their maximum is 8, is
(a) \( \frac{11}{40} \)
(b) \( \frac{3}{10} \)
(c) \( \frac{1}{40} \)
(d) None of the options
Answer: (a) \( \frac{11}{40} \)

Question. A man draws a card from a pack of 52 cards and then replaces it. After shuffling the pack, he again draws a card. This he repeats a number of times. The probability that he will draw a heart for the first time in the third draw is
(a) \( \frac{9}{64} \)
(b) \( \frac{27}{64} \)
(c) \( \frac{1}{4} \times \frac{{}^{39}C_2}{{}^{52}C_2} \)
(d) None of the options
Answer: (a) \( \frac{9}{64} \)

Question. A fair coin is tossed repeatedly. The probability of getting a result in the fifth toss different from those obtained in the first four tosses is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{32} \)
(c) \( \frac{31}{32} \)
(d) \( \frac{1}{16} \)
Answer: (d) \( \frac{1}{16} \)

Question. If the integers \( m \) and \( n \) are chosen at random between 1 and 100 then the probability that \( 7^m + 7^n \) is divisible by 5 is
(a) \( \frac{1}{4} \)
(b) \( \frac{1}{7} \)
(c) \( \frac{1}{5} \)
(d) \( \frac{1}{49} \)
Answer: (a) \( \frac{1}{4} \)

Question. It has been found that if A and B play a game 12 times, A wins 6 times, B wins 4 times and they draw twice. A and B take part in a series of 3 games. The probability that they will win alternately is
(a) \( \frac{5}{72} \)
(b) \( \frac{5}{36} \)
(c) \( \frac{19}{27} \)
(d) None of the options
Answer: (b) \( \frac{5}{36} \)