Given three identical boxes I, II and III each containing two coins. In the box I, both coins are gold coins, in box II, both are silver coins, and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?

Let E1 E2 E3 be events such that


E1 represents the selection of Box I


E2 represents Selection of Box II


And E3 represents the Selection of Box III


Let A denotes the event such that the coin drawn is of gold.


∵ there are 3 boxes-


∴ P(E1) = P(E2) = P(E3) = 1/3


Since, all boxes contains two coins.


And box I have both the coins gold.


Box II has only silver coins, and box III has one gold and one silver coin.


∴ P(A|E1) = P(drawing gold coin from box I) = 2/2 = 1


Similarly, P(A|E2) = 0


And P(A|E3) = 1/2


We need to find the probability of the event such that another coin in the box is also gold and this is only possible if the coin is drawn from the box I


i.e. we need to find P(E1|A).


From Baye’s theorem–



⇒ P(E1|A) =


∴ The probability that the other coin in the box is also of gold = 2/3


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