Three urns A,B and C contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.

Question

Philoid · Accepted Answer

Given:

Urn A has 6 red and 4 white balls

Urn B has 2 red and 6 white balls

Urn C has 1 red and 5 white balls

Let us assume U₁, U₂, U₃ and A be the events as follows:

U₁ = choosing Urn A

U₂ = choosing Urn B

U₃ = choosing Urn C

A = choosing red ball from urn

We know that each urn is most likely to choose. So, probability of choosing a urn will be same for every Urn.

⇒

The Probability of choosing balls from each Urn differs from Urn to Urn and the probabilities are as follows:

⇒ P(A|U₁) = P(Choosing red ball from Urn A)

⇒

⇒ P(A|U₂) = P(Choosing red ball from Urn B)

⇒

⇒ P(A|U₃) = P(Choosing red ball from Urn C)

⇒

Now we find

P(U₁|A) = P(The red ball is from Urn A)

Using Baye’s theorem:

⇒

∴ The required probability is .