Coin Flips Needed to Simulate a Fair Die

Each block of three flips independently succeeds with probability $6/8$.

The number of blocks needed is geometric; multiply its expectation by the number of flips per block.

1. 1. Find the block success probability

   Each three-bit block encodes a value in $\{0,\dots ,7\}$ uniformly, and only $6$ of the $8$ outcomes are accepted.

   $$p=\frac{6}{8}=\frac{3}{4}$$

2. 2. Expected number of blocks and flips

   The number of blocks needed is geometric with success probability $p$, so its expectation is $1/p$; each block costs three flips.

   $$E[\text{flips}]=3\times \frac{1}{p}=3\times \frac{4}{3}=4$$

Answer: 4