Conditional Probabilities of a fair coin

wanted to check this..

Define events A={first toss is a head}, B={three heads appear before two tails}

(the heads or tails do not have to be consecutive)

Find $P(B|A)$ and $P(B|A^c)$

I've got $P(B|A) = \frac{P(B \cap A)}{P(A)} = \frac{\frac{3}{32}}{\frac{1}{2}} = \frac{3}{16}$

and $P(B|A^c) =\frac{P(B \cap A^c)}{P(A^c)} = \frac{\frac{1}{32}}{\frac{1}{2}} = \frac{1}{16}$

:)