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.
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.
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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)
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⇒ P(A|U2) = P(Choosing red ball from Urn B)
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⇒ P(A|U3) = P(Choosing red ball from Urn C)
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Now we find
P(U1|A) = P(The red ball is from Urn A)
Using Baye’s theorem:
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∴ The required probability is .