Question

A coin is either fair or has two heads. You initially assign probabilities of 0.5 to each possibility. The coin is then tossed twice, with two heads appearing. Use bayes' theorem to work out the posterior probabilities of each possible outcome?Please answer the questions

Answer 1

Let p(F) be the likelihood of picking the fair coin. Let p(F’) be the likelihood of picking the biased coin. Absent any other information, it would be reasonable to assume that p(F)=p(F’)=1/2.

Define p(X) as the likelihood of getting two heads in a row. The fair coin has 4/16 chances of getting two heads in a row. So p(X|F)=4/16. By contrast, the biased coin has 9/16 chances of getting two heads in a row. So p(X|F’)=9/16.

By Bayes’ Rule,

p(F|X) = p(F,X) / p(X)

= p(X|F) * p(F) / p(X)

= p(X|F) * p(F) / [ p(X|F) * p(F) + p(X|F’) * p(F’) ]

= (4/16) * (1/2) / [ (4/16) * (1/2) + (9/16) * (1/2) ]

= 4/13