Suppose that we want to generate a random variable X that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability p. Consider the following procedure:

1. Flip the coin, and let 01, either heads or tails, be the result.

2. Flip the coin again, and let 02 be the result.

3. If 01 and 02 are the same, return to step 1.

4. If 02 is heads, set X = 0, otherwise set X = 1.

(a) Show that the random variable X generated by this procedure is equally likely to be either 0 or 1.

(b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different, and then sets X = 0 if the final flip is a head, and sets X = 1 if it is a tail?