Question: You are given biased coin. How would you find unbiased decision out of it?

Answer: First we assume that we have a single coin and that there are only 2 possible outcomes from tossing it: H and T.  By virtue of the problem definition, both H and T have a probability <> 0.5 but, due to our assumption, the sum of the probabilities must equal 1. So let p= the probability of getting a H. This implies that the probability of getting a T is (1-p).

On a single coin toss, the possible results are H or T with associated probabilities of p and (1-p). Hence it's a biased coin. Using single coin tosses is therefore not what we want to do.

Now consider 2 successive coin tosses using the biased coins. The possible outcomes of this are:

  1. HH
  2. HT
  3. TH
  4. TT

A coin toss is considered an independent random event. To calculate the probability of two independent random events we simply multiply the probability of each random event together. Given this, the probabilities of each of the four possible outcomes listed above are:

  • HH  :   p.p = p2
  • HT :    p(1-p) 
  • TH :    (1-p)p = p(1-p)
  • TT :    (1-p).(1-p) = (1-p)2

    From this it is clear that TH and HT events have the same probability so we need only consider these. So to answer the original question...

    Toss the biased coin twice. If the sequence is HH or TT ignore the result and start over again. If the result is HT call this a "head". If the result is a TH call the result a "tail". The likelihood of getting a "head" or "tail" is the same.