Average outcomes before they happen
Can an expected payoff be an amount you never actually receive?
Entering midway? A random variable assigns a number to each outcome. Probabilities must sum to one; a weighted mean multiplies values by their weights.
Start with a concrete question
An original fictional die game charges a fee and pays a prize on chosen winning faces of a fair six-sided die. No gambling or investment recommendation is implied.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict whether a positive maximum prize guarantees positive expected net payoff.
Change one thing. Inspect what happens.
- At defaults list both net outcomes and their probabilities.
- Calculate expectation before inspecting it; compare with either possible payoff.
- Raise prize and change winning faces separately; inspect variance as well as mean.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Two-outcome probability mass
Horizontal: Net credits. Vertical: Probability.
- 1. Losing outcome (points)
- 2. Winning outcome (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Net credits | Probability |
|---|---|---|
| Losing outcome | -3 | 0.8333 |
| Winning outcome | 9 | 0.1667 |
Follow the calculation
- A fair six-sided die has 1 winning faces: probability 1/6.
- Win net = 12 − 3 = 9; other net = −3.
- Expectation = 0.1666666667 × 9 + 0.8333333333 × (-3) = -1.
- Variance averages squared departures from that mean: 20 credits².
An expectation is a probability-weighted average, not necessarily a possible single outcome. Compare risk as well as average payoff.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Average outcomes before they happen
Recap for entering here
A random variable assigns a number to each outcome. Probabilities must sum to one; a weighted mean multiplies values by their weights.
The concrete question
An original fictional die game charges a fee and pays a prize on chosen winning faces of a fair six-sided die. No gambling or investment recommendation is implied.
Predict whether a positive maximum prize guarantees positive expected net payoff.
Work through the mechanism
The net random variable is prize minus fee on a win, and minus fee otherwise.
At defaults, net outcomes are 9 with probability 1/6 and −3 with probability 5/6. Expected net is 9/6−15/6=−1, though no play pays exactly −1.
Variance is the probability-weighted squared distance from the expected value. Here it is (1/6)×10²+(5/6)×(−2)²=20 credits².
Formula and symbols
E[X]=Σ pᵢxᵢ; Var(X)=Σ pᵢ(xᵢ−E[X])². X is net credits, pᵢ outcome probability, Σ means sum.
Test the idea
At defaults list both net outcomes and their probabilities.
Calculate expectation before inspecting it; compare with either possible payoff.
Raise prize and change winning faces separately; inspect variance as well as mean.
Assumptions and boundaries
A fair die and fixed payouts are assumed exactly.
Expectation describes an average under a model, not a guaranteed finite-run result or a complete risk preference.
Extend your investigation
High school: Create a complete outcome table for two dice.
College: Derive linearity of expectation and distinguish it from independence assumptions for variance.
Working professionals: Compare expected costs with downside constraints in a fictional operations decision.
Try a fresh case
A random payoff is 8 with probability 0.25 and −4 otherwise. What is its expectation?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Create a complete outcome table for two dice.
College investigation
Derive linearity of expectation and distinguish it from independence assumptions for variance.
Workplace investigation
Compare expected costs with downside constraints in a fictional operations decision.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.