The contents of the urns I,II,III are as follows: Urn I: 1 white, 2 black and 3 red balls Urn II: 2 white, 1 black and 1 red balls Urn III: 4 white, 5 black and 3 red balls. One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns I,II,III?

Question

Philoid · Accepted Answer

Given:

Urn I has 1 white, 2 black and 3 red balls

Urn II has 2 white, 1 black and 1 red balls

Urn III has 4 white, 5 black and 3 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 and 1 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 required balls from Urn 1)

⇒

⇒ P(A|U₂) = P(Choosing required balls 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)

P(U₂|A) = P(The chosen balls are from Urn2)

P(U₃|A) = P(The chosen balls are from Urn3)

Using Baye’s theorem:

⇒

∴ The required probabilities are .