Question

Probability of solving specific problem independently by A and B are respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

Answer 1

It is given that probability of A is P( A )= 1 2 , the probability of B is P( B )= 1 3 .

P( A∩B )=P( A )×P( B ) = 1 2 × 1 3 = 1 6

(i)

The probability for the problem can be written as,

Probability=P( A∪B ) =P( A )+P( B )−P( A∩B ) (1)

Here, the probability for the problem is solved is P. And it is given events A and B are independent so P( A∩B )= P( A )×P( B ).

Substitute the given values in (1) we get,

P= 1 2 + 1 3 − 1 6 = 2 3

Thus, the probability for the problem is solved is 2 3 .

(ii)

The probability can be calculated as,

Probability=P( A∩ B ′ )+P( B∩ A ′ ) =P( A )+P( B )−2P( A∩B ) (1)

Here, the probability for exactly one of them solves the problem is P and it is given events A and B are independent so P( A∩B )= P( A )×P( B ).

Substitute the given values in (1) we get,

P= 1 2 + 1 3 −2× 1 6 = 1 2

Thus, the probability for the problem is solved is 1 2 .