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# 3. Consider the following game. A bucket contains one black ball and n − 1 white...

3. Consider the following game. A bucket contains one black ball and n − 1 white balls. Two players take it in turn to draw balls from the bucket. On each turn, a player draws a ball from the bucket. If the player draws the black ball, then that player wins and the game ends. If the player draws a white ball, then the ball is returned to the bucket and the game continues until one player draws the black ball. (a) Alice and Bob decide to play this game with Alice being first to draw a ball from the bucket. What is the probability that Alice wins the game. (b) Bob thinks it is unfair that Alice always goes first in these games. Suppose they decide who goes first based on the toss a fair coin. If the result of the coin toss in heads, then Alice plays first. Otherwise, Bob plays first. What is the probability that Alice wins? (c) Assume that n = 10. What is the probability that 10 or more balls are drawn before there is a winner?

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