Given three identical boxes I, II and III, each containing two coins.

1.	Given three identical boxes I, II and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in the 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?
Answer» Let E₁, E₂ and E₃ be the events that boxes I, II and III are chosen, respectively Then P(E₁) = P(E₂) = P(E₃) = 1/3 Also, let A be the event that ‘the coin drawn is of gold’ Then P(A\|E₁) = P(a gold coin from bag I) = 2/2 =1 P(A\|E₂) = P(a gold coin from bag II) = 0 P(A\|E₃) = P(a gold coin from bag III) = 1/2 Now, the probability that the other coin in the box is of gold = the probability that gold coin is drawn from the box I. = P(E₁\|A) By Bayes' theorem, we know that P(E₁/A) = (P(E₁)P(A/E₁))/)(P(E₁)P(A/E₁) + P(E₂)P(A/E₂) + P(E₃)P(A/E₃)) = (1/3 x 1)/(1/3 x 1 + 1/3 x 0 + 1/3 x 1/2) = 2/3

Given three identical boxes I, II and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in the 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?

Answer»

Let E₁, E₂ and E₃ be the events that boxes I, II and III are chosen, respectively

Then P(E₁) = P(E₂) = P(E₃) = 1/3

Also, let A be the event that ‘the coin drawn is of gold’

Then P(A|E₁) = P(a gold coin from bag I) = 2/2 =1

P(A|E₂) = P(a gold coin from bag II) = 0

P(A|E₃) = P(a gold coin from bag III) = 1/2

Now, the probability that the other coin in the box is of gold = the probability that gold coin is drawn from the box I.

= P(E₁|A)

By Bayes' theorem, we know that

P(E₁/A) = (P(E₁)P(A/E₁))/)(P(E₁)P(A/E₁) + P(E₂)P(A/E₂) + P(E₃)P(A/E₃))

= (1/3 x 1)/(1/3 x 1 + 1/3 x 0 + 1/3 x 1/2) = 2/3