Expected value as a weighted average

Build an outcome table and keep the mean separate from a prediction of one result.

10 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Compute expected net value for a finite distribution and use linearity correctly.

Prerequisites: Net payoff and sunk costs · Chips, fractions and percentages

Weight mutually exclusive, exhaustive outcomes

For outcomes xᵢ with probabilities pᵢ, the expectation is Σpᵢxᵢ. The probabilities must cover the experiment and sum to one. If one scenario contains several distinct payoffs, split it or calculate its conditional mean first.

The mean need not itself be a possible result. A bet paying +30 or −10 with equal probability has mean +10, even though no single play pays +10. Repeated trials do not guarantee the mean after a specified finite number of attempts.

Linearity does not require independence

For any integrable payoffs X and Y, E[X+Y]=E[X]+E[Y], whether the two are independent, positively related or negatively related. Independence becomes important for other questions, such as multiplying event probabilities or adding variances.

Changing the scale of every payoff by a factor a changes the mean by a. Adding a fixed fee subtracts that fee from the mean. Those transformations are easy checks on arithmetic, but they do not say whether taking a risk is appropriate under a separate loss constraint.

Worked example

A model has a 40% chance of +60 chips and a 60% chance of −20 chips. A fixed 3-chip fee is charged for taking it.

Before the fee, the expected gain is 0.40×60+0.60×(−20)=24−12=12.
The fixed fee applies to every result, so subtract 3 once.
The resulting net outcomes are +57 and −23. Their weighted average is 22.8−13.8=9.

Expected net value is +9 chips after the fee. Paying the fee only in the losing branch would describe a different contract.

Common mistakes

Averaging payoffs without their probabilities.
Treating a mean as the most likely outcome.
Assuming expectation additivity requires independent trials.

Keep these ideas

Cover all outcomes and normalize weights.
The mean may not be a realizable individual result.
Linearity holds without independence.