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Probability theory is the branch of mathematics that deals with quantifying the likelihood of events. It's essential for understanding various random processes which underpin a wide variety of disciplines from physics to finance, and, as in this example, philanthropy efforts.

Probability is measured on a scale from 0 to 1, where 0 implies an impossible event and 1 represents an event that is certain to happen. Intermediate values are assigned to events based on their likelihood. For instance, Justine knows from her organization's experience that approximately 20% (\(0.20\) when expressed as a probability) of potential donors will agree to donate—a useful figure when planning outreach strategies.

The concept of expected value is pivotal in understanding the long-term average outcome of random events. It is essentially a weighted average of all possible outcomes, where the weights are their respective probabilities. To put it simply, it tells us what to expect on average over many trials of the same experiment.

In the case of Justine and her fundraising efforts, the expected value indicates the average number of potential donors needed to contact before securing a donation of \(100 or more. Calculating this requires understanding the geometric distribution, which models the number of trials needed to achieve a first success. Using the formula for the expected value of a geometrically distributed variable, \(1/p\), with \(p\) being the probability of a donor contributing \)100, Justine could expect to call 100 potential donors on average to achieve her goal.

• An independent event is one whose occurrence is not affected by any other events.
• For example, each potential donor's decision to give \(100 is independent of another's decision.

Understanding this concept is crucial when dealing with probabilities of combined events in the context of these charitable contributions. If two events are independent, as is the case with separate individuals deciding to donate, the probability of both events occurring is the product of their individual probabilities. In the exercise, the independent contribution decisions of potential donors are used to determine the overall probability that someone will donate \)100. This independent probability is 1%, found by multiplying the probability of agreeing to donate (20%) by the probability of donating $100 (5%).