Exercise 12.5: Choose the correct answer

Introduction to Probability Theory (Mathematics)

Choose the correct or most suitable answer from the given four alternatives

(1) Four persons are selected at random from a group of 3 men, 2 women and 4 children. The probability that exactly two of them are children is

(1) 3/4 

(2) 10/23 

(3) 1/2 

(4) 10/21

Ans: (4)

Solution

(2) A number is selected from the set { 1, 2, 3,..., 20 }. The probability that the selected number is divisible by 3 or 4 is 

(1) 2/5 

(2) 1/8 

(3) 1/2 

(4) 2/3 

Ans: (3)

Solution

(3) A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are 3/4 , 1/2 , 8/5 . The probability that the target is hit by A or B but not by C is

(1) 21 /64

(2) 7/32 

(3) 9/64 

(4) 7/8

Ans: (1)

Solution

(4) If A and B are any two events, then the probability that exactly one of them occur is

(3) P( A ) + P( B ) − P( A ∩ B ) 

(4) P ( A) + P ( B ) + 2 P ( A ∩ B)

Ans: (2)

Solution

(5) Let A and B be two events such that P = 1/6 , P ( A ∩ B) = 1/4 and P () = 1/4 . Then the events A and B are 

(1) Equally likely but not independent 

(2) Independent but not equally likely

 (3) Independent and equally likely 

(4) Mutually inclusive and dependent

Ans: (2)

Solution

(6) Two items are chosen from a lot containing twelve items of which four are defective, then the probability that at least one of the item is defective

(1) 19/33 

(2) 17/33 

(3) 23/33 

(4) 13/33

Ans: (1)

Solution

(7) A man has 3 fifty rupee notes, 4 hundred rupees notes and 6 five hundred rupees notes in his pocket. If 2 notes are taken at random, what are the odds in favour of both notes being of hundred rupee denomination?

(1) 1:12 

(2) 12:1 

(3) 13:1 

(4) 1:13

Ans: (1)

Solution

(8) A letter is taken at random from the letters of the word ‘ASSISTANT’ and another letter is taken at random from the letters of the word ‘STATISTICS’. The probability that the selected letters are the same is

(1) 7/45 

(2) 17/90 

(3) 29/90 

(4) 19/90

Ans: (4)

Solution

(9) A matrix is chosen at random from a set of all matrices of order 2, with elements 0 or 1 only. The probability that the determinant of the matrix chosen is non zero will be

(1) 3/16 

(2) 3/8 

(3) 1/4 

(4) 5/8

Ans: (2)

Solution

(10) A bag contains 5 white and 3 black balls. Five balls are drawn successively without replacement. The probability that they are alternately of different colours is 

(1) 3/14 

(2) 5/14 

(3) 1/14 

(4) 9/14

Ans: (3)

Solution

(11)  If A and B are two events such that A ⊂ B and P (B) ≠0, then which of the following is correct?

(1) P ( A / B) = P ( A) / P ( B)

(2) P ( A / B ) < P( A)

(3) P ( A / B ) ≥ P( A) 

(4) P ( A / B ) > P(B)

Ans: (3)

(12)  A bag contains 6 green, 2 white, and 7 black balls. If two balls are drawn simultaneously, then the probability that both are different colours is

(1) 68/105 

(2) 71/105 

(3) 64/105 

(4) 73/105 

Ans: (1)

Solution

(13) If X and Y be two events such that P ( X / Y ) = 1/2 , P ( Y / X ) = 1/3 and P ( X ∩ Y ) = 16 , then P ( X ∪Y ) is 

(1) 1/3 

(2) 2/5 

(3) 1/6 

(4) 2/3 

Ans: (4)

Solution

(14) An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. The probability that the second ball drawn is red will be

(1) 5/12 

(2) 1/2 

(3) 7/12 

(4) 1/4

Ans: (2)

Solution

(15) A number x is chosen at random from the first 100 natural numbers. Let A be the event of numbers which satisfies [(x − 10)(x − 50)] / [x − 30 ] ≥ 0 , then P ( A) is

(1) 0.20 

(2) 0.51 

(3) 0.71 

(4) 0.70

Ans: (3)

Solution

(16) If two events A and B are independent such that P ( A) = 0.35 and P ( A ∪ B) = 0.6 ,  then P (B) is 

(1) 5/13 

(2) 1/13 

(3) 4/13 

(4) 7/13 

Ans: (1)

Solution

(17) If two events A and B are such that P () = 3/10 and P ( A ∩  ) = 1/2 , then P ( A ∩ B) is

(1) 1/2 

(2) 1/3 

(3) 1/4 

(4) 1/5 

Ans: (4)

Solution

(18) If A and B are two events such that P (A)= 0.4, P ( B) = 0.8 and P ( B / A) = 0.6 , then  P ( âˆ© B) is 

(1) 0.96 

(2) 0.24 

(3) 0.56 

(4) 0.66

Ans: (3)

Solution

(19) There are three events A, B and C of which one and only one can happen. If the odds are 7 to 4 against A and 5 to 3 against B, then odds against C is

(1) 23: 65 

(2) 65: 23 

(3) 23: 88 

(4) 88: 23

Ans: (2)

Solution

(20) If a and b are chosen randomly from the set {1,2,3,4}with replacement, then the probability of the real roots of the equation x² + ax + b = 0 is

(1) 3/16 

(2) 5/16 

(3) 7/16 

(4) 11/16

Ans: (3)

Solution

(21) It is given that the events A and B are such that P( A) = 1/4 , P ( A / B) = 1/2 and P ( B / A) = 2/3 . Then P(B) is

(1) 1/6 

(2) 1/3 

(3) 2/3 

(4) 1/2

Ans: (2)

Solution

(22) In a certain college 4% of the boys and 1% of the girls are taller than 1.8 meter. Further 60% of the students are girls. If a student is selected at random and is taller than 1.8 meters, then the probability that the student is a girl is

(1) 2/11 

(2) 3/11 

(3) 5/11 

(4) 7/11 

Ans: (2)

Solution

(23) Ten coins are tossed. The probability of getting at least 8 heads is 

(1) 7/64 

(2) 7/32 

(3) 7/16 

(4) 7/128

Ans: (4)

Solution

(24) The probability of two events A and B are 0.3 and 0.6 respectively. The probability that both A and B occur simultaneously is 0.18. The probability that neither A nor B occurs is

(1) 0.1 

(2) 0.72 

(3) 0.42 

(4) 0.28

Ans: (4)

Solution

(25) If m is a number such that m ≤ 5, then the probability that quadratic equation 2x² + 2mx + m + 1 = 0 has real roots is 

(1) 1/5 

(2) 2/5 

(3) 3/5 

(4) 4/5 

Ans: (3)

Solution