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Question
There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?
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Solution
Let E1: The coin is head on both sides
E2: the coin is biased
E3: the coin is unbiased
E: The coin tossed shows the head
P(E1) = P(E2) = P(E3) = `1/3`
P(E|E1) = 1
P(E|E2) = 75% = `75/100 = 3/4`
P(E|E3) = `1/2`
Intended process
P(E1|E) = `(P(E_1) xx P(E|E_1))/(P(E_1) xx P(E|E_1) + P(E_2) xx P(E|E_2) + P(E_3) xx P(E|E_3)`
= `(1/3 xx 1)/(1/3 xx 1 + 1/3 xx 3/4 + 1/3 xx 1/2)`
= `1/(1 + 3/4 + 1/2)`
= `4/(4 + 3 + 2)`
= `4/9`
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