Just want to make sure I'm doing this problem correctly.

Suppose I have ten coins, 2 of which are heads on both sides, 3 that have a heads and a tails, and 5 that have tails on both sides. If I pick a coin at random and flip it, given that the side that lands face up is heads, what is the conditional probability that the other side is also heads?

So I'm looking for P(bottom side is heads | the top side is heads)

which is the conditional probability P(both sides are heads)/P(a coin lands on heads)

P(both sides are heads) = 2/10, since 2 of my coins are double-sided heads. P(a coin lands on heads) = 1*(2/10) + 0*(5/10) + (1/2)(3/10) = 7/20

So my answer would be (2/10)/(7/20) = 4/7. Is this the correct approach to this problem?

There is a more informal but fully correct way to find the answer without explicit appeal to the conditional probability "formula." There is a total of $7$ heads. Of these, $4$ belong to a two-headed coin. So the probability is $\frac{4}{7}$.