I am looking at the following problem about a coin toss experiment. I cannot understand the statements in the problem.
Problem statement is given below.
A drawer contains two coins. One is an unbiased coin, which when tossed, is equally likely to turn up heads or tails. The other is a biased coin, which will turn up heads with probability $p$ and tails with probability $1 − p$. One coin is selected (uniformly) at random from the drawer. Two experiments are performed:
a) The selected coin is tossed $n$ times. Given that the coin turns up heads $k$ times and tails $n − k$ times, what is the probability that the coin is biased?
b) The selected coin is tossed repeatedly until it turns up heads $k$ times. Given that the coin is tossed $n$ times in total, what is the probability that the coin is biased?
My confusion is that I cannot really figure out the difference between the two cases. Can you please help me understand the difference between the two cases?