A probability distribution models possible values of a random variable and assigns probabilities to them; its table, graph, or formula supports calculating and comparing event probabilities, expected values, and, when relevant, variance or standard deviation. Decision-making compares alternatives using a stated criterion such as expected payoff or expected cost, while recognizing that an expected value is a long-run average, not a guaranteed outcome, and that greater variability may represent greater risk. This scope excludes advanced utility theory, Bayesian decision analysis, and abstract distribution theory.
A probability distribution lists each possible outcome of a random variable and the probability of that outcome. To use it for a decision:
Example: You can choose either Plan A or Plan B for a repeated investment. Their possible payoffs are:
| Plan | Payoff | Probability |
|---|---|---|
| A | 0$ | |
| A | 100$ | |
| B | 60$ | |
| B | 80$ |
For Plan A:
For Plan B:
If the criterion is choose the plan with the greater expected payoff, choose Plan A, because
Plan A has more variability: it could produce either 0$100$40$.
Plan B’s payoffs are closer to its mean, and its standard deviation is 10$.
Therefore, Plan A has the greater expected payoff but also the greater risk. The expected payoff of 80$80$; it represents the long-run average payoff over many similar investments. If avoiding risk is more important than maximizing expected payoff, someone might prefer Plan B instead.
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