An urn contains red, green, and blue balls. Let r, g, b be the proportions of red

Chapter 2, Problem 57

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An urn contains red, green, and blue balls. Let r, g, b be the proportions of red, green, blue balls, respectively (r + g + b = 1). (a) Balls are drawn randomly with replacement. Find the probability that the first time a green ball is drawn is before the first time a blue ball is drawn. Hint: Explain how this relates to finding the probability that a draw is green, given that it is either green or blue. (b) Balls are drawn randomly without replacement. Find the probability that the first time a green ball is drawn is before the first time a blue ball is drawn. Is the answer the same or dierent than the answer in (a)? Hint: Imagine the balls all lined up, in the order in which they will be drawn. Note that where the red balls are standing in this line is irrelevant. (c) Generalize the result from (a) to the following setting. Independent trials are performed, and the outcome of each trial is classified as being exactly one of type 1, type 2, . . . , or type n, with probabilities p1, p2,...,pn, respectively. Find the probability that the first trial to result in type i comes before the first trial to result in type j, for i 6= j.