JEE Mathematics Probability MCQs Set F

Type – 1

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

Question. If the probability of A to fail in an examination is \( \frac{1}{5} \) and that of B is \( \frac{3}{10} \), then the probability that either A or B fails is
(a) \( \frac{1}{2} \)
(b) \( \frac{11}{25} \)
(c) \( \frac{19}{50} \)
(d) None of the options
Answer: (c) \( \frac{19}{50} \)

Question. A and B are two events where P(A) = 0.25 and P(B) = 0.5. The probability of both happening together is 0.14. The probability of both A and B not happening is
(a) 0.39
(b) 0.25
(c) 0.11
(d) None of the options
Answer: (a) 0.39

Question. Three faces of an ordinary dice are yellow, two faces are red and one face is blue. The dice is tossed 3 times. The probability that yellow, red and blue faces appear in the first, second and third tosses respectively is
(a) \( \frac{1}{36} \)
(b) \( \frac{1}{6} \)
(c) \( \frac{1}{30} \)
(d) None of the options
Answer: (a) \( \frac{1}{36} \)

Question. India play two matches each with West Indies and Australia. In any match the probabilities of India getting 0, 1 and 2 points are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points is
(a) 0.0875
(b) \( \frac{1}{16} \)
(c) 0.1125
(d) None of the options
Answer: (a) 0.0875

Question. Let A and B be two independent events such that \( P(A) = \frac{1}{5} \) and \( P(A \cup B) = \frac{7}{10} \). Then \( P(\bar{B}) \) is equal to
(a) \( \frac{3}{8} \)
(b) \( \frac{2}{7} \)
(c) \( \frac{7}{9} \)
(d) None of the options
Answer: (a) \( \frac{3}{8} \)

Question. Let A and B be two independent events such that their probabilities are \( \frac{3}{10} \) and \( \frac{2}{5} \). The probability of exactly one of the events happening is
(a) \( \frac{23}{50} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{31}{50} \)
(d) None of the options
Answer: (a) \( \frac{23}{50} \)

Question. The probability that at least one of the events A and B occurs is \( \frac{3}{5} \). If A and B occur simultaneously with probability \( \frac{1}{5} \), then P(A') + P(B') is
(a) \( \frac{2}{5} \)
(b) \( \frac{4}{5} \)
(c) \( \frac{6}{5} \)
(d) \( \frac{7}{5} \)
Answer: (c) \( \frac{6}{5} \)

Question. A, B, C are three events for which P(A) = 0.6, P(B) = 0.4, P(C) = 0.5, P(A ∪ B) = 0.8, P(A ∩ C) = 0.3 and P(A ∪ B ∩ C) = 0.2. If P(A ∪ B ∪ C) ≥ 0.85 then the interval of values of P(B ∩ C) is
(a) [0.2, 0.35]
(b) [0.55, 0.7]
(c) [0.2, 0.55]
(d) None of the options
Answer: (a) [0.2, 0.35]

Question. A coin is tossed 2n times. The chance that the number of times one gets head is not equal to the number of times one gets tail is
(a) \( \frac{(n)!}{(n!)^2} \cdot \left( \frac{1}{2} \right)^{2n} \)
(b) \( 1 - \frac{(2n)!}{(n!)^2} \)
(c) \( 1 - \frac{(2n)!}{(n!)^2} \cdot \frac{1}{4^n} \)
(d) None of the options
Answer: (c) \( 1 - \frac{(2n)!}{(n!)^2} \cdot \frac{1}{4^n} \)

Question. A coin is tossed n times. The probability of getting at least one head is greater than that of getting at least two tails by 5/32. Then n is
(a) 5
(b) 10
(c) 15
(d) None of the options
Answer: (a) 5

Question. A coin is tossed 7 times. Each time a man calls head. The probability that he wins the toss on more occasions is
(a) \( \frac{1}{4} \)
(b) \( \frac{5}{8} \)
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (c) \( \frac{1}{2} \)

Question. A bag contains 14 balls of two colours, the number of balls of each colour being the same. 7 balls are drawn at random one by one. The ball in hand is returned to the bag before each new draw. If the probability that at least 3 balls of each colour are drawn is p then
(a) \( p > \frac{1}{2} \)
(b) \( p = \frac{1}{2} \)
(c) \( p < 1 \)
(d) \( p < \frac{1}{2} \)
Answer: (a) \( p > \frac{1}{2} \)

Question. From a box containing 20 tickets of value 1 to 20, four tickets are drawn one by one. After each draw, the ticket is replaced. The probability that the largest value of tickets drawn is 15 is
(a) \( \left( \frac{3}{4} \right)^4 \)
(b) \( \frac{27}{320} \)
(c) \( \frac{27}{1280} \)
(d) None of the options
Answer: (b) \( \frac{27}{320} \)

Question. A dice is thrown 2n + 1 times, \( n \in N \). The probability that faces with even numbers show odd number of times is
(a) \( \frac{2n + 1}{4n + 3} \)
(b) less than \( \frac{1}{2} \)
(c) greater than \( \frac{1}{2} \)
(d) None of the options
Answer: (d) None of the options

Question. 6 ordinary dice are rolled. The probability that at least half of them will show at least 3 is
(a) \( 41 \times \frac{2^4}{3^6} \)
(b) \( \frac{2^4}{3^6} \)
(c) \( 20 \times \frac{2^4}{3^6} \)
(d) None of the options
Answer: (a) \( 41 \times \frac{2^4}{3^6} \)

Question. An ordinary dice is rolled a certain number of times. The probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times. Then the probability of getting an odd number an odd number of times is
(a) \( \frac{1}{32} \)
(b) \( \frac{5}{16} \)
(c) \( \frac{1}{2} \)
(d) None of the options
Answer: (c) \( \frac{1}{2} \)

Question. A card is drawn from a pack. The card is replaced and the pack is reshuffled. If this is done six times, the probability that 2 hearts, 2 diamonds and 2 black cards are drawn is
(a) \( 90 \cdot \left( \frac{1}{4} \right)^6 \)
(b) \( \frac{45}{2} \cdot \left( \frac{3}{4} \right)^4 \)
(c) \( \frac{90}{2^{10}} \)
(d) None of the options
Answer: (c) \( \frac{90}{2^{10}} \)

Question. A man firing at a distant target has 10% chance of hitting the target in one shot. The number of times he must fire at the target to have about 50% chance of hitting the target is
(a) 11
(b) 9
(c) 7
(d) 5
Answer: (c) 7

Question. There are four machines and it is known that exactly two of them are faulty. They are tested one by one in a random order till both the faulty machines are identified. Then the probability that only two tests will be required is
(a) \( \frac{1}{3} \)
(b) \( \frac{1}{6} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{1}{4} \)
Answer: (b) \( \frac{1}{6} \)

Question. Let A = {2, 3, 4, ..., 20, 21}. A number is chosen at random from the set A and it is found to be a prime number. The probability that it is more than 10 is
(a) \( \frac{9}{10} \)
(b) \( \frac{1}{10} \)
(c) \( \frac{1}{5} \)
(d) None of the options
Answer: (c) \( \frac{1}{5} \)

Question. All the spades are taken out from a pack of cards. From these cards, cards are drawn one by one without replacement till the ace of spades comes. The probability that the ace comes in the 4th draw is
(a) \( \frac{1}{13} \)
(b) \( \frac{12}{13} \)
(c) \( \frac{4}{13} \)
(d) None of the options
Answer: (a) \( \frac{1}{13} \)

Question. A point is selected at random from the interior of a circle. The probability that the point is closer to the centre than the boundary of the circle is
(a) \( \frac{3}{4} \)
(b) \( \frac{1}{2} \)
(c) \( \frac{1}{4} \)
(d) None of the options
Answer: (c) \( \frac{1}{4} \)

Question. A, B and C are contesting the election for the post of secretary of a club which does not allow ladies to become members. The probabilities of A, B and C winning the election are respectively \( \frac{4}{15} \), \( \frac{1}{3} \) and \( \frac{2}{5} \). The probabilities of introducing the clause of admitting lady members to the club by A, B, and C are 0.6, 0.7 and 0.5 respectively. The probability that ladies will be taken as members in the club after the election is
(a) \( \frac{26}{45} \)
(b) \( \frac{5}{9} \)
(c) \( \frac{19}{45} \)
(d) None of the options
Answer: (a) \( \frac{26}{45} \)

Question. There are 4 white and 3 black balls in a box. In another box there are 3 white and 4 black balls. An unbiased dice is rolled. If it shows a number less than or equal to 3 then a ball is drawn from the first box but if it shows a number more than 3 then a ball is drawn from the second box. If the ball drawn is black then the probability that the ball was drawn from the first box is
(a) \( \frac{1}{2} \)
(b) \( \frac{6}{7} \)
(c) \( \frac{4}{7} \)
(d) \( \frac{3}{7} \)
Answer: (d) \( \frac{3}{7} \)

Type 2

Choose the correct options. One or more options may be correct.

Question. The probabilities that a student passes in mathematics, physics and chemistry are m, p and c respectively. Of these subjects, a student has a 75% chance of passing in at least one, a 50% chance of passing in at least two, and a 40% chance of passing in exactly two subjects. Which of the following relations are true?
(a) \( p + m + c = \frac{19}{20} \)
(b) \( p + m + c = \frac{27}{20} \)
(c) \( pmc = \frac{1}{10} \)
(d) \( pmc = \frac{1}{4} \)
Answer: (b) \( p + m + c = \frac{27}{20} \)
(c) \( pmc = \frac{1}{10} \)

Question. If E and F are two events with P(E) ≤ P(F) > 0 then
(a) occurrence of E ⇒ occurrence of F
(b) occurrence of F ⇒ occurrence of E
(c) nonoccurrence of E ⇒ nonoccurrence of F
(d) none of the above implications hold
Answer: (d) none of the above implications hold

Question. If A and B are two events such that P(A ∪ B) > 3/4 and 1/8 ≤ P(A ∩ B) ≤ 3/8 then
(a) P(A) + P(B) ≤ 11/8
(b) P(A) . P(B) < 3/8
(c) P(A) + P(B) ≥ 7/8
(d) None of the options
Answer: (a) P(A) + P(B) ≤ 11/8
(c) P(A) + P(B) ≥ 7/8

Question. If \( \bar{E} \) and \( \bar{F} \) are the complementary events of the events E and F respectively then
(a) P(E/F) + P(\( \bar{E} \)/F) = 1
(b) P(E/F) + P(E/\( \bar{F} \)) = 1
(c) P(\( \bar{E} \)/F) + P(E/\( \bar{F} \)) = 1
(d) P(E/\( \bar{F} \)) + P(\( \bar{E} \)/\( \bar{F} \)) = 1
Answer: (a) P(E/F) + P(\( \bar{E} \)/F) = 1
(d) P(E/\( \bar{F} \)) + P(\( \bar{E} \)/\( \bar{F} \)) = 1

Question. Given that x ∈ [0,1] and y ∈ [0,1]. Let A be the event of (x, y) satisfying y² < x and B be the event of (x, y) satisfying x² < y. Then
(a) P(A ∩ B) = 1/3
(b) A, B are exhaustive
(c) A, B are mutually exclusive
(d) A, B are independent
Answer: (a) P(A ∩ B) = 1/3

Question. Let A and B be two events such that P(A ∩ B) = 1/3, P(A ∪ B) = 1 and P(\( \bar{A} \)) = 1/2. Then
(a) A, B are independent
(b) A, B are mutually exclusive
(c) P(A) = P(B)
(d) P(B) P(A)
Answer: (a) A, B are independent

Question. The probability that exactly one of the independent events A and B occurs is
(a) P(A) + P(B) - 2P(A ∩ B)
(b) P(A) + P(B) - P(A ∩ B)
(c) P(\( \bar{A} \) ∩ B) + P(A ∩ \( \bar{B} \))
(d) None of the options
Answer: (a) P(A) + P(B) - 2P(A ∩ B)
(c) P(\( \bar{A} \) ∩ B) + P(A ∩ \( \bar{B} \))

Question. If A and B are independent events such that 0 < P(A) < 1, 0 < P(B) < 1 then
(a) A, B are mutually exclusive
(b) A and \( \bar{B} \) are independent
(c) \( \bar{A} \), \( \bar{B} \) are independent
(d) P(A/B) + P(\( \bar{A} \)/B) = 1
Answer: (b) A and \( \bar{B} \) are independent
(c) \( \bar{A} \), \( \bar{B} \) are independent
(d) P(A/B) + P(\( \bar{A} \)/B) = 1

Question. For any two events A and B
(a) P(A ∩ B) ≥ P(A) + P(B) – 1
(b) P(A ∩ B) ≥ P(A) + P(B)
(c) P(A ∩ B) = P(A) + P(B) - P(A ∪ B)
(d) P(A ∩ B) = P(A) + P(B) + P(A ∪ B)
Answer: (a) P(A ∩ B) ≥ P(A) + P(B) – 1
(c) P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

Question. A coin is tossed repeatedly. A and B call alternately for winning a prize of Rs 30. One who calls correctly first wins the prize. A starts the call. Then the expectation of
(a) A is Rs 10
(b) B is Rs 10
(c) A is Rs 20
(d) B is Rs 20
Answer: (b) B is Rs 10 (c) A is Rs 20