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Question

Given 3 identical boxes 1,2, & 3. Each containing 2 coins. In box 1, both are gold coins & in box 2 both are silver coins and in box 3 there is 1 silver and 1 gold coin. A person chooses a box at random and takes out the coin. If the coin is gold then what is the probability that it is coming from the first box?

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Solution

Let E1,E2,E3 be the event that boxes 1, 2 and 3 are chosen respectively.

Then,

P(E1)=P(E2)=P(E3)=13

Let A be the event that the coin drawn is of gold.

Then, P(A|E1)=22=1

P(A|E2)=0

P(A|E3)=12

Probability that other gold coin is drawn from Bag 1 = P(E1|A)

By Baye's theorem,

P(E1|A)=P(E1)P(A|E1)P(E1)P(A|E1)+P(E2)P(A|E2)

=13×113×1+13×0+13×12=23

Hence, required probability is 23.

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