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The expected value is an important concept in probability and statistics. It helps us determine the average outcome of a probabilistic event over multiple trials. To find the expected value of a gamble, multiply each outcome by its probability and then add all of these values together. For our problem:- The gamble offers \(1000\) dollars with a probability of \(25\% = 0.25\).- It also offers \(100\) dollars with a probability of \(75\% = 0.75\).The expected value (EV) can be calculated as follows:\[\text{EV} = (1000 \times 0.25) + (100 \times 0.75)\]Calculating gives:\[250 + 75 = 325\]This expected value represents a way to quantify the average amount you can anticipate winning per gamble if you were to play many times.

Probability measures the likelihood of a particular outcome occurring. In our exercise, we examine the probabilities associated with two potential outcomes of a gamble.- There is a \(25\%\) probability that the outcome will be \(1000\) dollars.- Similarly, there is a \(75\%\) probability of the outcome being \(100\) dollars.These percentages add up to \(100\%\), indicating that we've accounted for all possible outcomes. Understanding probability is essential in determining the expected value because it tells us how frequently we can expect each outcome to occur relative to others. It is crucial to grasp that while individual outcomes are uncertain, their average behavior in the long run can be predicted using probabilities.

A certain payment, unlike a gamble, removes uncertainty. In our problem, the certain payment is \(325\) dollars. This means the individual is guaranteed to receive \(325\) dollars without facing any risk. For risk-averse individuals, who generally prefer safe and secure outcomes, a certain payment is often more attractive if its value matches or is near the expected value of a risky option. Risk aversion plays a key role here.In situations where the expected value of a gamble equals the certain payment, like in this exercise, risk-averse individuals would choose the certain payment since it offers the same value without any uncertainty. Understanding this behavior is fundamental in economics and finance, especially when making decisions under uncertainty.