Question

a a fair coin a fair coin is one that has probability 1 over 2 of coming up heads when flapped what is the probability that a fair coin will come up tails n time in a row find the probability that a coin is comes after head for the first time after an even number of coins flips?​

Answer 1

HELLO FRIENDS

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It can also be alternatively answered by Bayes Rule, which says that prior odds times likelihood ratio is the posterior odds.

So we can first calculate the odds ratio of: (the fair coin given it was a head in the first toss) to (the biased coin given it was heads in the first toss).

We know: P(Fair) = 1/2 , P(Biased) = 1/2

We also know: P(Head|Fair) = 1/2 , P(Head|Biased) = 3/4

So let's first calculate the odds ratio after the first toss

i.e. P(Fair|Head)/P(Bias|Head) = [P(Head|Fair)/P(Head|Biased)] x [P(Fair)/P(Biased)]

Odds ratio will be: [(1/2)/(3/4)] x [(1/2)/(1/2)] = 2/3

We then update this odds ratio after the second head using the Bayes rule that prior odds times likelihood ratio is the posterior odds.

2/3 will become the prior odds ratio to calculate the posterior odds.

New Posterior odds = Prior odds x [P(Head|Fair)/P(Head|Biased)]

New Posterior odds = (2/3) x [(1/2)/(3/4)] = 4/9

New posterior odds is actually: P(Fair|Head,Head)/P(Bias|Head,Head)

P(Fair|Head,Head)/P(Bias|Head,Head) = 4/9

From Cox's axioms P(Fair|Head,Head) + P(Bias|Head,Head) = 1

so we get P(Fair|Head,Head) = 4/13 i.e. probability of picking a fair coin given it was heads both the times.

HOPE IT WILL BE HELPFUL TO YOU