Question

An urn contains 4 white balls and 6 red balls. A second urn contains 7 white balls and 3 red balls. An urn is selected, and the probability of selecting the first urn is 0.1. A ball is drawn from the selected urn and replaced. Then another ball is drawn and replaced from the same urn. If both balls are white, what are the following probabilities? (Round your answers to three decimal places.)

- (a) the probability that the urn selected was the first one

- (b) the probability that the urn selected was the second one

Answer 1

U1 : Event of selecting the first urn

U2 : Event of selecting the second urn

P(U1) = 0.1

P(U2) = 1-0.1=0.9

W : Event drawing two white balls with replacement (One white is drawn and replaced and then second white ball is drawn)

P(W|U1) : Probability of drawing a white ball from the Urn 1 replaced back and drawing a second white ball  

Probability of drawing a white ball(first) from the Urn 1 = Number of white balls in the Urn1 / total number of balls in the urn1= 4/10=0.4

The ball is replaced back, and the second ball is drawn

Probability of the drawing a white ball(second) from the Urn 1 = Number of white balls in the Urn1 / total number of balls in the Urn 1 = 4/10=0.4

P(W|U1) : Probability of drawing a white ball from the Urn 1 replaced back and drawing a second white ball =0.4 x 0.4 =0.16

P(W|U2) : Probability of drawing a white ball from the Urn 2 replaced back and drawing a second white ball  

Probability of drawing a white ball(first) from the Urn 2 = Number of white balls in the Urn2 / total number of balls in the urn 2 = 7/10=0.7

The ball is replaced back, and the second ball is drawn

Probability of the drawing a white ball(second) from the Urn 1 = Number of white balls in the Urn2 / total number of balls in the urn 2= 7/10=0.7

P(W|U2) : Probability of drawing a white ball from the Urn 2 replaced back and drawing a second white ball = 0.7 x 0.7 = 0.49

(a)

If both balls are white, probability that the urn selected was the first one = P(U1|W)

By using Bayes theorem

P(U1)P(W|U1) = 0.1 x 0.16= 0.016

P(U2)P(W|U2) = 0.9 x 0.49=0.441

P(U1)P(W|U1) + P(U2)P(W|U2) = 0.016+0.441= 0.457

If both balls are white, probability that the urn selected was the first one = P(U1|W) = 0.035

If both balls are white, probability that the urn selected was the first one = 0.035

(b)

If both balls are white, probability that the urn selected was the second one = P(U2|W)

By using Bayes theorem

P(U2)P(W|U2) = 0.9 x 0.49=0.441

P(U1)P(W|U1) = 0.1 x 0.16= 0.016

P(U1)P(W|U1) + P(U2)P(W|U2) = 0.016+0.441= 0.457

If both balls are white, probability that the urn selected was the first one = P(U2|W) = 0.965

If both balls are white, probability that the urn selected was the second one = 0.965