1. Suppose that you roll a standard 6-sided die and then based on the outcome ch

Question

1. Suppose that you roll a standard 6-sided die and then based on the outcome choose a ball from one of four urns. If you roll a 1, you'll pick from urn A, a 2 you'll pick from urn B, a 3 you'll pick from urn C, and a 4, 5, or 6 you'll pick from urn D. If urn A contains 2 red and 8 white balls, urn B 6 red and 4 white balls, urn C 5 red and 5 white balls, and urn D 4 red and 6 white balls, what is the probability that a ball was drawn from urn B given that the ball drawn is red? 0.10 0.16 0.20 0.24

Explanation / Answer

P(urn A) = P(rolling a 1 with a fair die) = 1/6

P(urn B) = P(rolling a 2 with a fair die) = 1/6

P(urn C) = P(rolling a 3 with a fair die) = 1/6

P(urn D) = P(rolling a 4, 5, or 6 with a fair die) = 3/6 = 1/2

P(red| urn A) = 2/(2+8) = 1/5

P(red| urn B) = 6/(6+4) = 6/10 = 3/5

P(red| urn C) = 5/(5+5) = 5/10 = 1/2

P(red|urn D) = 4/(4+6) = 2/5

Then P(urn B|red) = P(urn B and red)/P(red) =

P(urn B and red)/P(red and (urn A or urn B or urn C or urn D)) =

P(urn B and red)/((P(red and urn A)+P(red and urn B) + P(red and urn C) + P(red and urn D)) =

P(urn B)P(red|urn B)/(P(urn A)P(red|urn A)+P(urn B)P(red|urn B)+P(urn C)P(red|urn C)+P(urn D)P(red|urn D)) =

1/6*3/5/(1/6*1/5+1/6*3/5+1/6*1/2+1/2*2/5) =

3/30/(1/30+3/30+1/12+2/10) =

6/60/(2/60+6/60+5/60+12/60) =

6/60/(25/60) =

6/25 = 0.24

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