A two-outcome cash-flow example separates the mathematical average from the median. Neither number replaces the contract’s loss distribution.

What the source establishes

NIST’s binomial model assigns probabilities to two-outcome trials. It does not make the expected value the most likely realised result. A payoff calculation must separately define the money received in each outcome and the price paid. NIST/SEMATECH — binomial probability model ↗

Consider a purely illustrative contract costing 4 units and paying 100 units with probability 5%, otherwise zero, with no fees. Expected gross receipt is 5 units and expected net result is positive 1. Yet the net result is minus 4 with probability 95%, so the median single-contract result is minus 4. These are hypothetical inputs, not a quoted product or forecast.

What the distinction changes

A positive mathematical expectation can coexist with a high probability of loss on a single trial. That is a property of the specified distribution, not a contradiction. Describing only the expected result would hide most of the outcome probability in this example.

Repeated trials introduce further questions about dependence, changing probabilities, cash constraints and costs. The average of a model is not a path through time. A sales claim that turns expected return into the usual customer outcome is making an additional assertion.

Questions for the next review

Check gross versus net payout, fees, scenario probabilities and the evidence behind them. Report the loss probability and possible cash outcomes alongside the average. Do not replace an unverified success estimate with extra decimal places and call the result factual performance.