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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. What is the probability that it lands head up?
Solution
Let E1 ≡ the event that first fair coin is selected
E2 ≡ the event that second fair coin is selected
E3 ≡ the event that two-headed coin is selected
H ≡ the event that head turns up
Then P(E1) = P(E2) = P(E3) = `1/3`
and `"P"("H"/"E"_1) = 1/2, "P"("H"/"E"_2) = 1/2, "P"("H"/"E"_3)` = 1
Head will turn up if anyone of E1 ∩ H, E2 ∩ H, E3 ∩ H, occurs.
These events are mutually exclusive
∴ P(H) = P(E1 ∩ H) + P(E2 ∩ H) + P(E3 ∩ H)
= `"P"("E"_1)*"P"("H"/"E"_1) + "P"("E"_2)*"P"("H"/"E"_2) + "P"("E"_3)*"P"("H"/"E"_3)`
= `1/3*1/2 + 1/3* 1/2 + 1/3*1` ...[E1, E2, E3 are equally likely]
= `2/3`.
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