Question

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 $0_{1}$, either heads or tails, be the result. 2. Flip the coin again, and let $0_{2}$ be the result. 3. If $0_{1}$ and $0_{2}$ are the same, return to step 1 . 4. If $0_{2}$ 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?

          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 $0_{1}$, either heads or tails, be the result.
2. Flip the coin again, and let $0_{2}$ be the result.
3. If $0_{1}$ and $0_{2}$ are the same, return to step 1 .
4. If $0_{2}$ 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?

Added by Sherry A.

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