Suppose you are given a biased coin for which the probability of getting a head is $p (0<p<1)$.Discuss how you will select one of two individuals at random using the biased coin.
As listed in the commments, the canonical way to generate an unbiased coin flip for a biased coin, as first proposed by Von Neumann, is as follows:
Toss the coin twice.
If the result is either both heads of both tails, discard and toss the coin twice again.
- Otherwise, (if they are different), record the outcome of the first coin toss.
Understanding why this works relies on the observation that if you have a biased coin that comes up heads with probability $p$, and if you flip the coin twice, then:
- The probability of HH is: $p^2$
- The probability of HT is: $p(1-p)$
- The probability of TH is: $(1-p)p$
- The probability of TT is: $(1-p)^2.$