Consider a roulette wheel consisting of 38 numbers 1 through...

Consider a roulette wheel consisting of 38 numbers 1 through $36,0,$ and double $0 .$ If Smith always bets that the outcome will be one of the numbers 1 through $12,$ what is the probability that
(a) Smith will lose his first 5 bets;
(b) his first win will occur on his fourth bet?

Key Concepts

Geometric Distribution

The geometric distribution models the number of independent trials needed to achieve the first success. It is particularly relevant when determining the probability that the first win happens on a specific trial, such as the fourth bet, by calculating the probability of consecutive failures followed by a success.

Compound Event Probability

This concept refers to finding the probability that several independent events occur together. When considering multiple bets, the overall probability of a series of wins and losses is computed by multiplying the probabilities of the individual events. This multiplication rule is key when determining the probability of sequences, such as consecutive losses in multiple bets.

Roulette Wheel Sample Space

This concept involves understanding the total number of possible outcomes in an experiment. In the case of the roulette wheel described, each spin can result in one of 38 numbers. Recognizing the complete sample space is essential for determining individual event probabilities and for setting up the probability calculations.

Independent Events

In probability theory, independent events are those whose outcomes do not affect each other. Each bet made on the roulette wheel is independent of previous bets, meaning the probability of a win or loss remains constant from one spin to the next. This principle is crucial when calculating the probability of multiple sequential outcomes.

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