Given a list of coin tosses with 100,000 outcomes, suppose you know that they were generated by either a fair or a biased coin with a 51% chance of heads.
How do you determine which coin it was generated by?
Suppose that you start with no opinion about which coin you
have.
Then we have two hypotheses:
A= The event that the coin is fair.
Let the complement of A be the event that the coin is biased.
If B is the result of a single flip,
Bayes theorem says:
I'm still confused as to how I should compute these probabilities? If I know the first flip is tails, what does that say about the probability of A given B?? Thank you!
"Start with no opinion about which coin you have" might be intended to be interpreted as "start with $P(A) = P(A^c) = 1/2$."
As you flip each coin, you can update your belief. Let $B$ be the event that the first flip is tails. Then $P(B \mid A) = 1/2$ and $P(B \mid A^c) = 0.49$. You can now use Bayes's formula to compute $P(A \mid B)$.
As suggested by the caption to your screenshot, this is now your "current estimate of $P(A)$" which you will use if you are going to use Bayes's formula to deal with the next flip, for instance.