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 E_{1} ≡ the event that first fair coin is selected

E_{2} ≡ the event that second fair coin is selected

E_{3} ≡ the event that two-headed coin is selected

H ≡ the event that head turns up

Then P(E_{1}) = P(E_{2}) = P(E_{3}) = `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 E_{1} ∩ H, E_{2} ∩ H, E_{3} ∩ H, occurs.

These events are mutually exclusive

∴ P(H) = P(E_{1} ∩ H) + P(E_{2} ∩ H) + P(E_{3} ∩ 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` ...[E_{1}, E_{2}, E_{3} are equally likely]

= `2/3`.