Please solve this question Justify your answer Suppose you h

Please solve this question. Justify your answer.

Suppose you have a biased coin with P(heads) notequalto 0.5. How could you use this coin to simulate a fair coin?

Solution

Let consider that H be the event that the coin flip comes up with heads and T be the event that the coin flip comes up with tails.

since coin is biased we do not know the probability of getting heads but let’s assume that the variable p = P(H)

Now do the mini experiment in which we flip the same coin twice.

Let assume that,

HT be the event that we will get heads and then tails in two consecutive flips,

TH be the event that we will get tails and then heads in two consecutive flips.

HH be the event that we will get two heads in a row, and

TT be the event that we will get two tails in a row.

we can say that sample space for the experiment is then = {HH,HT,TH,TT}

we know that P(H) = p , we can write the probabilities of the events in our sample space as :

P(HH) = P(First toss H)P(Second toss H|First toss H)

But as the coind is biased and first toss does not remember the first toss

P(HH) = P(H)P(H) = p²

simillarly we can say that,

P(HT) = P(First toss H)P(Second toss T|First toss H) = P(H)(1P(H)) = p(1 p)

P(T H) = P(First toss T)P(Second toss H|First toss T) = (1P(H))P(H) = p(1 p),

P(T T) = P(First toss T)P(Second toss T|First toss T) = (1P(H))(1P(H)) = (1 p)²

We can see that the probability P(HT) and the probability P(TH) are equal, i.e., p(1 p).

so we can say that by symmetry, these two probabilities must be the same since the coin doesn’t know what it came up before.

Hence we can simulate a fair coin using the following process.

Toss the coin twice. If the outcome comes out to be heads both times or tails both times, throw away the result and repeat the whole process again. Otherwise, if the outcome is HT, we return “heads”, and if the outcome is TH, we return “tails”.