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# Probability Analysis : Roulette

1. Roulette is played at a table similar to the one in Figure 3.37. A wheel with the numbers 1 through 36 (evenly distributed with the colors red and black) and two green numbers 0 and 00 rotates in a shallow bowl with a curved wall. A small ball is spun on the inside of the wall and drops into a pocket corresponding to one of the numbers. Players may make 11 different types of belts by placing chips on different areas of the table. These include bets on a single number, two adjacent numbers, a row of three numbers, a block of four numbers, two adjacent rows of six numbers, and the five number combinations of 0, 00, 1, 2, and 3; bets on the numbers 1 - 18 or 19 - 36; the first, second, or third group of 12 numbers; a column of 12 numbers; even or odd; and red or black. Payoffs differ by bet. For instance, a single-number bet pays 35 to 1 in it wins; a three-number bet pays 11 to 1; a column bet pays 2 to 1; and a color bet pays even money. Define the following events: C1 = column 1 number, C2 = column 2 number, C3 = column 3 number, O = odd number, E = even number, G = green number, F12 = first 12 numbers, S12 = second 12 numbers and T12 = third 12 numbers.

Probability Rules and Calculations

Rule 1. The probability associated with any outcome must be between 0 and 1.
Rule 2. The sum of the probabilities over all possible outcomes must be 1.0.
Rule 3. The probability of any event is the sum of the probabilities of the outcomes that compose that event.
Rule 4. If events A and B are mutually exclusive, then P(Aor B) = P(A) + P(B).
Rule 5. If two events A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) - P(A and B).

a. Find P(G or O), P(O or F12), P(C1 or C3), P(E and F12), P(E or F12), P(S12 and T12), P(O or C2).

#### Solution Preview

We know that the total number of outcomes is 38. They are 0, 00, 1-36. From the condition, we have the following table.
C1: 3n+1 or 1, 4, 7, ..., 34, totally 12 outcomes.
C2: 3n+2 or 2, 5, 8, ..., 35, totally 12 ...

#### Solution Summary

Roulette probabilities are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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