Module 11 of 15

Expected Value

Weighing outcomes by how likely they are, and knowing when that is not enough.

AMathematical Foundation

The expected value of an uncertain prospect with outcomes x₁, …, xₖ and probabilities p₁, …, pₖ (summing to 1) is E[X] = Σ pᵢxᵢ. It is the long-run average payoff if the prospect were repeated many times. Expected value is linear and easy to compare, but it ignores the spread of outcomes. Variance Σ pᵢ(xᵢ − E[X])² and the worst case describe risk; decision-makers who cannot absorb large losses, or who face a one-off irreversible choice, may reasonably prefer a lower expected value with less risk.

E[X]=\sum_{i=1}^{k} p_i x_i,\qquad \operatorname{Var}(X)=\sum_{i=1}^{k} p_i\,(x_i-E[X])^2

In words: The expected value is the sum of probability times payoff; the variance is the probability-weighted sum of squared deviations from the expected value.

$p_i$: probability of outcome i (all p_i ≥ 0 and sum to 1)
$x_i$: payoff of outcome i, in a common unit

Assumptions

All outcomes can be expressed on one scale (money, wellbeing, lives).
The probabilities are known or credibly estimated.
The decision-maker is risk-neutral, or the payoffs already reflect attitudes to risk (utilities).

BIntuitive Explanation

A lottery ticket that costs $2 and pays $1 million with probability 1 in 10 million has an expected value of 10 cents: a bad deal on average. An insurance policy has a negative expected value for the buyer too, yet buying it is often sensible, because it removes a catastrophic outcome. Expected value is the right starting point and the wrong stopping point.

CConnection to Debate

Debaters constantly make implicit expected-value claims: “even a small chance of a huge benefit justifies the policy”, or “the risk is unlikely, so ignore it”. Making the calculation explicit forces each side to defend its probabilities and magnitudes. It also exposes where expected value is not the right standard: when harms fall on people who did not consent (a child in a family vlog), when outcomes are irreversible, or when rights constrain what may be traded for benefits.

DReal-World Application

Outcome-contingent tuition as a pricing problem
Competing interpretation
Income-share agreements, a real form of outcome-contingent tuition, price expected repayment: Purdue's Back a Boiler programme (2016–2022) varied borrowing limits and terms by major, and students paid a percentage of salary for about ten years. A provider's expected revenue is the sum, across students, of each repayment weighted by its probability, so it has an incentive to favour programmes with high predicted earnings. Regulators have also scrutinised such contracts: the CFPB found in 2021 that one provider's ISAs were loans and that its prepayment penalties were unlawful.
Sources
- Purdue University Suspends Income-Share Agreements, Its Loan AlternativeNadia Tamez-Robledo, EdSurge, 2022-06-24
- CFPB Takes Action Against Student Lender for Misleading Borrowers About Income Share AgreementsConsumer Financial Protection Bureau, 2021-09-07
Mandatory prenuptial agreements
Competing interpretation
The expected benefit of a prenuptial agreement is a probability-weighted sum: the chance of divorce times the difference the agreement makes to the outcome, minus its costs. Population figures such as CDC's 2.4 divorces per 1,000 population in 2023 are not that probability for a particular couple, and the difference an agreement makes depends on the assets involved and on the default legal regime. The numbers in this module's interactive are hypothetical.
Sources
- FastStats: Marriage and DivorceU.S. Centers for Disease Control and Prevention, National Center for Health Statistics, 2023

These examples describe real policies and research. The numbers in the interactive below and in the practice questions are hypothetical teaching data unless a question says otherwise.

EInteractive Visualization

Edit probabilities and payoffs for two options and compare expected value, spread, worst case and probability of loss.

Illustrative example — hypothetical data.Hypothetical payoffs in thousands of dollars of lifetime net earnings. Change probabilities and payoffs to test the decision.

Option A: high-variance path

P(Breakthrough)5%

Payoff

P(Modest success)25%

Payoff

P(Failure)

70.0% (the remainder)

Payoff

Option B: conventional path

P(Good outcome)30%

Payoff

P(Typical outcome)60%

Payoff

P(Poor outcome)

10.0% (the remainder)

Payoff

Option A: high-variance path

Expected value

16.0

Standard deviation

91.8

Worst case

-20

Probability of a loss

70%

Option B: conventional path

Expected value

28.5

Standard deviation

13.0

Worst case

Probability of a loss

E[X]=\sum_i p_i\,x_i,\qquad \sigma=\sqrt{\sum_i p_i\,(x_i-E[X])^2}

In words: The expected value is the sum of each probability times its payoff; the standard deviation is the square root of the probability-weighted squared distance from the expected value.

Option B: conventional path has the higher expected value (28.5 vs 16.0).

Expected value assumes the probabilities are known, the payoffs capture everything that matters, and the choice is repeated often enough for averages to apply. A one-off, irreversible decision about a child’s life or a family’s savings may violate all three.

FDebate Practice

GLimitations

Expected value cannot tell you how to compare incommensurable outcomes (money vs. dignity), whose consent matters, or how to treat tiny probabilities of catastrophic outcomes. Its precision is only as good as the probability estimates, which in policy debates are often guesses. Use it to structure disagreement, not to end it.

AMathematical Foundation

E[X]=\sum_{i=1}^{k} p_i x_i,\qquad \operatorname{Var}(X)=\sum_{i=1}^{k} p_i\,(x_i-E[X])^2

In words: The expected value is the sum of probability times payoff; the variance is the probability-weighted sum of squared deviations from the expected value.

$p_i$: probability of outcome i (all p_i ≥ 0 and sum to 1)
$x_i$: payoff of outcome i, in a common unit

Assumptions

All outcomes can be expressed on one scale (money, wellbeing, lives).
The probabilities are known or credibly estimated.
The decision-maker is risk-neutral, or the payoffs already reflect attitudes to risk (utilities).

BIntuitive Explanation

CConnection to Debate

DReal-World Application

Outcome-contingent tuition as a pricing problem
Competing interpretation
Income-share agreements, a real form of outcome-contingent tuition, price expected repayment: Purdue's Back a Boiler programme (2016–2022) varied borrowing limits and terms by major, and students paid a percentage of salary for about ten years. A provider's expected revenue is the sum, across students, of each repayment weighted by its probability, so it has an incentive to favour programmes with high predicted earnings. Regulators have also scrutinised such contracts: the CFPB found in 2021 that one provider's ISAs were loans and that its prepayment penalties were unlawful.
Sources
- Purdue University Suspends Income-Share Agreements, Its Loan AlternativeNadia Tamez-Robledo, EdSurge, 2022-06-24
- CFPB Takes Action Against Student Lender for Misleading Borrowers About Income Share AgreementsConsumer Financial Protection Bureau, 2021-09-07
Mandatory prenuptial agreements
Competing interpretation
The expected benefit of a prenuptial agreement is a probability-weighted sum: the chance of divorce times the difference the agreement makes to the outcome, minus its costs. Population figures such as CDC's 2.4 divorces per 1,000 population in 2023 are not that probability for a particular couple, and the difference an agreement makes depends on the assets involved and on the default legal regime. The numbers in this module's interactive are hypothetical.
Sources
- FastStats: Marriage and DivorceU.S. Centers for Disease Control and Prevention, National Center for Health Statistics, 2023

These examples describe real policies and research. The numbers in the interactive below and in the practice questions are hypothetical teaching data unless a question says otherwise.

EInteractive Visualization

Edit probabilities and payoffs for two options and compare expected value, spread, worst case and probability of loss.

Illustrative example — hypothetical data.Hypothetical payoffs in thousands of dollars of lifetime net earnings. Change probabilities and payoffs to test the decision.

Option A: high-variance path

P(Breakthrough)5%

Payoff

P(Modest success)25%

Payoff

P(Failure)

70.0% (the remainder)

Payoff

Option B: conventional path

P(Good outcome)30%

Payoff

P(Typical outcome)60%

Payoff

P(Poor outcome)

10.0% (the remainder)

Payoff

Option A: high-variance path

Expected value

16.0

Standard deviation

91.8

Worst case

-20

Probability of a loss

70%

Option B: conventional path

Expected value

28.5

Standard deviation

13.0

Worst case

Probability of a loss

E[X]=\sum_i p_i\,x_i,\qquad \sigma=\sqrt{\sum_i p_i\,(x_i-E[X])^2}

In words: The expected value is the sum of each probability times its payoff; the standard deviation is the square root of the probability-weighted squared distance from the expected value.

Option B: conventional path has the higher expected value (28.5 vs 16.0).

GLimitations