People face uncertain situations. Uncertainty arises in situations where future outcomes are unknown and influenced by chance or external factors. A college student may get a high-paying job as soon as they graduate in the future or remain unemployed for a long time. 

Another example of uncertainty is a college basketball team playing the final game of a championship. The team may either win the final game of the championship and earn the prize money or lose and earn nothing.

Outcomes are the potential results of an uncertain event. For example, in a basketball championship, the outcomes are either winning the game and the championship or losing both. 

Payoffs represent the value associated with each outcome. Assume the team receives $10,000 if it wins and $6,000 if it loses. So, the payoffs in this example are $10,000 and $6,000.

Probability quantifies the likelihood of different outcomes in uncertain situations. For example, it is assumed that the college team has a 0.5 probability of winning the final game and a 0.5 probability of losing the final game.

Expected value is a useful tool for evaluating uncertain situations. It represents the average payoff across all possible outcomes, weighted by their probabilities. It is calculated by summing the products of each payoff and its associated probability. It is expressed as:

Expected Value = (P₁×X₁) + (P₂×X₂)

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

The expected income, which is the estimate of the college team’s expected earnings, is

(0.5×$10,000)+(0.5×$6,000) = $8,000

This analysis is helpful in decision-making under uncertainty.