If the probability of heads for coin 1 is 0.4, and the probability of heads is 0.7 for coin 2, and assuming either coin is selected with equal probability:

a. What is the probability that 7 of 10 flips is heads?

I did part a simply enough, $0.5[\left( \begin{matrix} 10\\ 7\end{matrix} \right) \*0.4^7\*0.6^3 + \left( \begin{matrix} 10\\ 7\end{matrix} \right) \*0.7^7\*0.3^3]$

Part b is is a bit confusing to me:
What is the conditional probability that 7 of 10 flips is heads given that the 1st flip is heads?

I have event A: the first flip is heads, and event B, 7 of the 10 flips are heads:

So the $P(B|A)=\dfrac {P(A \cap B)} {P(A)}$. $P(A) = .5(0.4+0.7)=0.55$ How do I take the dependence of event B on event A into account when figuring this out? Our professor assured us that the answer in the solutions manual is wrong for part b.