American roulette is a game in which a wheel turns on a...

American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered 1–36, of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on three consecutive spins.

American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered 1–36, of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets.
(a) Find the probability of landing in the number 00 pocket.
(b) Find the probability of landing in a red pocket.
(c) Find the probability of landing in a green pocket or a black pocket.
(d) Find the probability of landing in the number 14 pocket on two consecutive spins.
(e) Find the probability of landing in a red pocket on three consecutive spins.

Key Concepts

Uniform Probability Distribution

This concept assumes that all outcomes in the sample space are equally likely to occur. In games such as American roulette, each pocket (whether numbered or colored) has an equal chance of being chosen, which is fundamental when calculating probabilities by using simple ratios of favorable outcomes to the total number of outcomes.

Sample Space

The sample space is the complete set of all possible outcomes in an experiment. For American roulette, it consists of all 38 pockets on the wheel. Understanding the sample space is essential as it forms the denominator when determining the probability of any event occurring.

Favorable Outcomes and Probability Calculation

Probability is calculated by comparing the number of favorable outcomes to the total number of outcomes in the sample space. This means that for any event, such as landing on a specific number or color, one must count how many outcomes satisfy that event and divide by the total number of outcomes available.

Addition Rule for Mutually Exclusive Events

When two events cannot happen at the same time, they are considered mutually exclusive, and the probability of either event occurring is the sum of their individual probabilities. This rule is useful when calculating probabilities of events like landing on either a green pocket or a black pocket.

Independent Events and the Multiplication Rule

Independent events are those whose outcomes do not affect one another. In the context of consecutive spins in roulette, each spin is independent. The probability of two or more independent events occurring in sequence is the product of their individual probabilities, which is critical when calculating outcomes over multiple spins.