5. A jar contains 4 pink marbles and 6 blue marbles. (a) For Experiment X, three

Question

5. A jar contains 4 pink marbles and 6 blue marbles.

(a) For Experiment X, three marbles are taken out of the jar without replacement. Given that there is at least one blue marble in the sample, what is the probability that the sample contains marbles of both colors?

(b) For Experiment Y, a trial consists of taking out a marble at random and if the marble is pink, it is not replaced and if the marble is blue, it is replaced. Determine the probability that the 2nd pink marble is observed on the 4th trial.

Explanation / Answer

5. (a) P(sample contains marbles of both colors | there is at least one blue marble in the sample)

= P(sample contains marbles of both colors) / P(there is at least one blue marble in the sample)

= [1 - P(all pink marbles) - P(all blue marbles)] / [1 - P(all pink marbles)]

= [1 - (4C3/10C3) - (6C3/10C3)] / [1 - 4C3/10C3]

= [1 - 0.0333 - 0.1667] / [1 - 0.0333]

= 0.8/0.9667

= 0.8276

(b) Let P denote taking a pink marble and B denote taking a blue marble

P(2nd pink marble on 4th trial) = P(PBBP) + P(BPBP) + P(BBPP)

= (4/10 x 6/9 x 6/9 x3/9) + (6/10 x 4/10 x 6/9 x 3/9) + (6/10 x 6/10 x 4/10 x 3/9)

= 0.0593 + 0.0533 + 0.048

= 0.1606

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