Here is the layout of the question:
3 coins, 2 are fair but the third is biased with a probability $4/7$ for heads. You randomly choose one of these coins, tossing it 10 times, and obtain 5 heads 5 tails. You then choose a different coin, again toss 10 times, and obtain 4 heads 6 tails. What is the probability that the remaining coin is the biased coin?
I have a feeling Bayes theorem could be helpful with this problem but I am unsure how and where to apply it. Is this just the multiplication of two conditional probabilities (namely $P(\text{first is fair}|5H,5T)$ and $P(\text{first is fair}|4H,6T)$)? Thanks:)