Expected winnings in roulette. In the popular casino game of roulette, you can bet on whet...

Expected winnings in roulette. In the popular casino game of roulette, you can bet on whether the ball will fall in an arc on the wheel colored red, black, or green. You showed (Exercise 3.133, p. 156) that the probability of a red outcome is 18/38, that of a black outcome is 18/38, and that of a green outcome is 2/38. Suppose you make a $5 bet on red. Find your expected net winnings for this single bet. Interpret the result.

Winning at roulette.Roulette is a very popular game in many American casinos. In Roulette, a ball spins on a circular wheel that is divided into 38 arcs of equal length, bearing the numbers 00, 0, 1, 2, … , 35, 36. The number of the arc on which the ball stops is the outcome of one play of the game. The numbers are also colored in the manner shown in the following table.

Players may place bets on the table in a variety of ways, including bets on odd, even, red, black, high, low, etc. Consider the following events:

A: {The outcome is an odd number (00 and 0 are considered neither odd nor even.)}

B: {The outcome is a black number.}

C: {The outcome is a low number (1–18).}

a. Define the event A∩ B as a specific set of sample points.

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b. Define the event A∩ B as a specific set of sample points.

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c. Find P(A), P(B), P(A ∩ B), P(A∩ B), and P(C) by summing the probabilities of the appropriate sample points.

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d. Define the event A∩ B∩ C as a specific set of sample points.

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e. Use the additive rule to find P(A∩ B). Are events A and B mutually exclusive? Why?

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f. Find P(A∩ B∩ C) by summing the probabilities of the sample points given in part d.

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d. Define the event A∩ B∩ C as a specific set of sample points.

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g. Define the event (A∩ B∩ C) as a specific set of sample points.

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h. Find P(A∩ B∩ C) by summing the probabilities of the sample points given in part g.

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g. Define the event (A∩ B∩ C) as a specific set of sample points.