Find probability the remaining coin is biased???

Question

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:)

Answer 1

Perhaps, first try to answer these:

• If the first coin is biased, what is the probaility of the observed outcomes for the two coins?
• If the second coin is biased, what is the probaility of the observed outcomes for the two coins?
• If the third coin is biased, what is the probaility of the observed outcomes for the two coins?