The probability of solving the problem independently by A and B are given as

1

2

and

1

3

respectively.
i.e. P (A) =

1

2

, P (B) =

1

3

,
∴ P (A ∩ B) = P (A) . P (B)
[Since the events corresponding to A and B are independent]
=

1

2

×

1

3

=

1

6

(i) Probability that the problem is solved
= P (A ∪ B)
= P (A) + P (B) - P (A ∩ B)
=

1

2

+

1

3

−

1

6

=

3+2−1

6

=

4

6

=

2

3

Thus, the probability that the problem is solved is

2

3

(ii) Probability that exactly one of them solves the problem
= P (A - B) + P (B - A)
= [P (A) - P (A ∩ B) + [P (B) - P (A ∩ B)]
= (

1

2

−

1

6

) + (

1

3

−

1

6

)
=

3−1+2−1

6

=

3

5

=

1

2

Thus, the probability that exactly one of them solves the problem is

1

2