Question

A problem in Mathematics is given to three students whose chances of solving it are 1/2,1/3,1/4 respectively. If the events of their solving the problem are independent then the probability that the problem will be solved, is

(a) 1/4

(b) 1/3

(c) 1/2

(d) 3/4

Answer 1

Correct option is (d) 3/4

Method 1:

Let A,B,C be the respective events of solving the problem. Then, P(A) = 1/2,P(B)= 1/3 and P(C) = 1/4 Here, A,B,C are independent events. Problem is solved if at least one of them solves the problem.

Required probability is = P(A ∪ B ∪ C) = 1 - P(\(\bar{A}\))P(\(\bar{B}\))P(\(\bar{C}\))

= 1 - (1 - 1/2)(1 - 1/3)(1 - 1/4)

= 1 - 1/4

= 3/4

Method 2:

The problem will be solved if one or more of them can solve the problem. The probability is P(\(\overline{ABC}\)) + P(\(\bar{A}B\bar{C}\)) + P(\(\overline{AB}C\)) +  P(\({A}B\bar{C}\)) +  P(\({A}\bar{B}{C}\)) +  P(\(\bar{A}{B}{C}\)) + P(ABC)

= 1/2,2/3,3/4 + 1/2,1/3,3/4 + 1/2,2/3,1/4 + 1/2,2/3,3/4 + 1/2,2/3,1/4 + 1/2,1/3,1/4 + 1/2,1/3,1/4

= 3/4

Method 3:

Let us think quantitively. Let us assume that there are 100 questions given to A. A solves 1/2 x 100 questions then remaining 50 questions is given to B and B solves 50 x 1/3 = 16.67 questions . Remaining 50 x 2/3 questions is given to C and C solves 50 x 2/3 x 1/4 = 8.33 questions.

Therefore, number of questions solved is 50 + 16.67 + 8.33 = 75. So, required probability is 75/100 = 3/4.