Question

Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is 2-headed. If you take one out, toss it and get a head, what is the probability that it was a fair coin?

Answer 1

Let E₁ = Event that the coin is fair

E₂ = Event that the coin is 2-headed

And H = Event that the tossed coin gets head.

P(E₁)  `1/2`

P(E₂)  `1/2`

`"P"("H"/"E"_1) = 1/2`

`"P"("H"/"E"_2)` = 1

∴ Using Bayes’ Theorem, we get

`"P"("E"_1/"H") = ("P"("E"_1)*"P"("H"/"E"_1))/("P"("E"_1)*"P"("H"/"E"_1) + "P"("E"_2)*"P"("H"/"E"_2))`

= `(1/2*1/2)/(1/2*1/2 + 1/2*1)`

= `(1/4)/(1/4 + 1/2)`

= `(1/4)/(3/4)`

= `1/3`

Hence the required probability is `1/3`.