Let E1 : ‘A solves the problem’ and E2 : ‘B solves the problem’, then P(E1)=

1

2

and P(E2)=

1

3

.
= Probability that the problem is solved by the least one of A and B
=P(E1)P(E2)+P(E1)P(E2)+P(E1)P(E2)
=(

1

2

×

2

3

)+(

1

2

×

1

3

)+

1

2

×

1

3

=

2

3

(ii) P(Exactly one of them solves the problem) = P(A solves the problem and B does not) + P (B solves the problem and A does not)
P(E1∩E2)+P(E2∩E1) =P(E1)P(E2)+P(E2)P(E1) ( ∵E1&E2 are independent)
=P(E1)(1−P(E2))+P(E2)(1−P(E1))
=(

1

2

×

2

3

)+(

1

3

×

1

2

)=

1

2