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Question
A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed. If happens to be head, what is the probability that it is the two-headed coin?
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Solution
Let event A: Fair coin is tossed,
event B: Fake coin is tossed and
event H: Head occur.
Clearly, a fair coin has one head.
∴ Probability that head occur under the condition that the fair coin is tossed = `"P"("H"/"A") = 1/2`.
Fake coin has two heads.
∴ Probability that head occur under the condition that the fake coin is tossed
= `"P"("H"/"B")` = 1
n(A) = 2, n(B) = 1, n(S) = 3
∴ P(A) = `("n"("A"))/("n"("S"))=2/3`, P(B) = `("n"("B"))/("n"("S"))=1/3`
Required probability = `"P"("B"/"H")`
By Baye’s theorem
`"P"("B"/"H") = ("P"("B") "P"("H"/"B"))/("P"("H"))`
= `(1/3 xx 1)/(2/3)`
= `1/2`
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