Actually the asnwer is 3/4.

Lets go through the conditional probabilities.

If the first child is named William and the second is NOT: the prob. of the second child being a boy is 1/2.

If the second child is named William and the first is NOT: the prob. of the second child being a boy is 1.

If both children are named William: the prob. of the second child being a boy is 1.

Now, assuming equal probs. of the first, second, or both children being named William, the total probability of the second child being a boy is (1/3)(1/2)+(1/3)(1)+(1/3)(1)= 5/6

the second ans should be 1/3