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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?
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Solution
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`.
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