Three urns contains 2 white and 3 black balls; 3 white and 2 black balls; 4 white and 1 black balls respectively. One ball is drawn from an urn chosen at random and it was found to be white. Find the probability that it was drawn from the first urn.

Question

Philoid · Accepted Answer

Given:

Urn I has 2 white and 3 black balls

Urn II has 3 white and 2 black balls

Urn III has 4 white, 1 black red balls

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

U₁ = choosing Urn I

U₂ = choosing Urn II

U₃ = choosing Urn III

A = choosing 1 white ball from an 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 white ball from Urn 1)

⇒

⇒ P(A|U₂) = P(Choosing white ball from Urn 2)

⇒

⇒ P(A|U₃) = P(Choosing required balls from Urn 3)

⇒

Now we find

P(U₁|A) = P(The chosen balls are from Urn1)

Using Baye’s theorem:

⇒

∴ The required probabilities is .