Consider a hypothetical example where John is evaluating a job offer from a company. If the company performs well, John will earn an annual income of  $81,000; if it performs poorly, he will earn $49,000. Each outcome is equally likely, with a probability of 0.5. These two outcomes are mutually exclusive, meaning only one can occur and their probabilities sum to 1. The amounts of $81,000 and $49,000 represent the payoffs associated with each outcome.

John's expected income is the average amount he expects to earn, considering all possible outcomes and their probabilities. 

Expected Income = (P₁ × X₁) + (P₂ ×  X₂)

where P₁ and P₂ are the probabilities of the two outcomes, and X₁ and X₂ are their corresponding payoffs.

For John,

• If the company performs well, his income will be $81,000 with a probability of 0.5.
• If the company does not perform well, his income will be $49,000 with a probability of 0.5.

Thus, John’s expected income is (0.5 × $81,000) + (0.5 × $49,000) = $65,000

This analysis is helpful in decision-making under uncertainty. However, the expected utility should be calculated for further analysis because income alone does not capture how John values different income levels. To do this, the utility values corresponding to all possible income levels are required. It is assumed that John experiences diminishing marginal utility of income, meaning additional income leads to progressively smaller increases in satisfaction. The utility values for the different income levels are shown in the following table:

Table: Utility-Income Relationship