Gambling Suppose a casino offers a gambling game involving a European roulette wheel, which has 37 slots numbered 0 through 36. The gambler's chance of winning depends on the number of chips he has when the wheel is spun. If the number of chips is a multiple of three, he wins one chip if the roulette wheel comes up 1, 2, or 3; otherwise he loses one chip. If the number of chips is not a multiple of 3 he wins one chip if the roulette wheel is any number between 1 and 28, inclusive; otherwise he loses one chip. Source: Math Horizons.
(a) Find the average chance of winning if one assumes that the number of chips the gambler possesses is a multiple of three one-third of the time.
(b) In fact, the number of chips the gambler possesses is not a multiple of three one-third of the time. To see this, let the number of chips the player has be modeled by a Markov chain with states 0, 1, and 2, based on the remainder when the player's chips are divided by three. Find the transition matrix for this Markov chain.
(c) Find the probability of being in state 0, 1, and 2 in the long run.
(d) Based on the long run probabilities from part (c), find the gambler's average chance of winning.
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Finite Mathematics (11th Edition)
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