(a) A candy vending machine is out of order. The probability...

(a) A candy vending machine is out of order. The probability that you get a candy bar (with or without the return of your quarter) is $\frac{1}{2}$, the probability that you get your quarter back (with or without candy) is $\frac{1}{3}$, and the probability that you get both the candy and your money back is $\frac{1}{12}$. What is the probability that you get nothing at all? Sugrestion: Sketch a geometric diagram similar to Figure $3.1$, indicate regions representing the various possibilities and their probabilities; then set up a four-point samplc space and the associated probabilities of the points. (b) Suppose you. put another quarter into the candy vending machine of part (a). Set up the 16-point sample space corresponding to the possible results of your two attempts to bu? a candy bar, and find the probability that you get two candy bars (and no money back): that you get no candy and lose both quarters; that you just get your money back both times.

(a) A candy vending machine is out of order. The probability that you get a candy bar (with or without the return of your quarter) is $\frac{1}{2}$, the probability that you get your quarter back (with or without candy) is $\frac{1}{3}$, and the probability that you get both the candy and your money back is $\frac{1}{12}$. What is the probability that you get nothing at all? Sugrestion: Sketch a geometric diagram similar to Figure $3.1$, indicate regions representing the various possibilities and their probabilities; then set up a four-point samplc space and the associated probabilities of the points.
(b) Suppose you. put another quarter into the candy vending machine of part (a). Set up the 16-point sample space corresponding to the possible results of your two attempts to bu? a candy bar, and find the probability that you get two candy bars (and no money back): that you get no candy and lose both quarters; that you just get your money back both times.

Key Concepts

Sample Space Construction

In probability, the sample space represents the set of all possible outcomes of an experiment. For the candy vending machine problem, constructing a sample space helps by detailing every possible result, such as getting candy, getting the quarter back, getting both, or getting nothing. For multiple trials (such as two attempts), the sample space is expanded to include all ordered pairs of outcomes, which is essential for analyzing compound events.

Events and Their Probabilities

An event is a subset of the sample space to which a probability is assigned. In this problem, we deal with events like obtaining a candy bar, receiving a refund, or the combination of both. Accurately assigning and calculating these probabilities, particularly through the intersections and unions of events (e.g., probability of both candy and quarter back), is fundamental in understanding and solving such problems.

Inclusion?Exclusion Principle

The Inclusion-Exclusion Principle is used to calculate the probability of the union of two events by accounting for the overlap between them. In this case, knowing the probability of getting candy, the probability of getting a refund, and the probability of both, one can find the probability of getting at least one of the outcomes. This technique is crucial to avoid double counting and to determine the probability of complementary events (such as obtaining nothing).

Independent Events in Compound Experiments

When multiple trials are performed, such as using the vending machine twice, each trial can be considered an independent event if the outcome of one does not affect the other. The overall probability of compound events, like getting two candy bars without a refund or getting the quarter back both times, is found by multiplying the individual probabilities. Understanding this principle is key when analyzing multi-step experiments.

Geometric/Diagrammatic Representation

A geometric or diagrammatic representation, such as a Venn diagram or probability tree, helps illustrate the relationships and overlaps between different events. It provides a visual method to organize and compute the probabilities of various outcomes, aiding in the understanding of how individual probabilities interact within the overall sample space.

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