Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he reaches into his pocket and discovers that the box picked is empty. If it is assumed that each of the matchboxes originally contained matches, what is the probability that there are exactly matches in the other box?
Let denote the event that the man discovers the matchbox in his right pocket is empty and there are matches in the matchbox in his left pocket. This event occurs only if the th choice of the matchbox in his right pocket is made at the trial.
Since it is equally likely that the matchbox found to be empty is in the left pocket, the desired probability is