Chapter 3, Problem 3.63P

Suppose that we want to generate the outcome of the flip of a fair coin, but that all we have at our disposal is a biased coin that lands on heads with some unknown probability p that need not be equal to $\frac{1}{2}$. Consider the following procedure for accomplishing our task:

1. Flip the coin.

2. Flip the coin again.

3. If both flips land on heads or both land on tails, return to step 1.

4. Let the result of the last flip be the result of the experiment.

a. Show that the result is equally likely to be either heads or tails.

b. Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?