Sports economics probability and expectations

Probability notation

The mathematical language for describing how likely an event is to occur, such as the probability that a team wins or a player makes a free throw.

P(A and B) = P(A) × P(B) (independent events)

The probability that both events occur when they are independent. For example, the chance that a player makes two consecutive free throws if each shot is independent.

P(not A) = 1 − P(A)

The probability that an event does not happen. For example, if a player has a 0.8 chance of scoring, there is a 0.2 chance they miss.

Odds and odds against

A way to express probability as a ratio. Odds of 4 to 1 against mean the event happens one time out of five, or a probability of 0.2.

Probability of discrete variables

Calculating the likelihood of specific countable outcomes, such as the probability of scoring exactly 3 goals in a match.

Probability of continuous variables

Calculating the likelihood of outcomes within a range for variables that can take on any value, like a player’s shooting percentage being between 45% and 50%.

Expected value (E[x|y])

The predicted average outcome of a variable x given another condition y. In sports, it could represent the expected points scored given a certain defensive strategy.

E[x + y] = E[x] + E[y]

The expected value of a sum equals the sum of expected values. For example, the expected total points from two players equals the sum of their individual expected points.

Conditional probability

The probability that an event occurs given that another event has already happened. For example, the probability a team wins given they’re leading at halftime.

Conditional expectation

The expected average outcome given specific conditions, such as expected goals given the number of shots taken.

Using probabilities in decision making

Teams and analysts use probabilities and expectations to make strategic choices, like when to attempt a fourth-down conversion or which player to sign based on expected performance value.

Law of total probability

States that the total probability of an event can be found by summing conditional probabilities across all possible conditions. In sports, it might describe total win probability as the sum of win probabilities given different score margins.

Relationship to conditional probabilities

Conditional probabilities form the building blocks of the law of total probability since each conditional outcome contributes to the overall likelihood of an event.

Field goal percentage example

Probability concepts apply to shooting statistics; for example, a player’s field goal percentage reflects the probability of making a shot. Adjusted stats (like expected field goal percentage) account for shot distance or defensive pressure to better reflect true performance.

Binomial distribution

A probability model for counting the number of successes in a fixed number of independent attempts, such as the number of made free throws out of 10 attempts.

Use of binomial distribution

Helps estimate consistency and predict outcomes across repeated trials, such as predicting how many games a team will win out of a season’s schedule.

Z-score

A standardized score showing how far a value is from the mean in standard deviations. For example, a player’s z-score in scoring shows how their points per game compare to the league average.

Using z-scores to compare performances

Allows comparison of players across different categories or eras by adjusting for league averages and variability.