How many tries before the average settles?

It depends on the spread of the outcomes rather than on their average, and it is usually far more than intuition suggests. The uncertainty around an observed average shrinks with the square root of the number of tries, so cutting it in half needs four times as many. For a bet whose outcomes are close together a few dozen tries give a stable picture; for one that pays thirty-five times its stake once in thirty-seven, thousands of spins can still end well away from the expected value in either direction, which is exactly why a run of good luck proves nothing.

Can outcomes be things other than money?

Yes — any quantity that can be added and multiplied works, which is what makes the idea travel so far. Expected delivery times, expected number of defects in a batch, expected repair costs across a fleet and expected calls per hour at a helpdesk are all the same calculation with different units. The one requirement is that adding the quantity has to mean something: expected temperature across three rooms is a legitimate average, while an expected value computed over categories with numeric labels — postcodes, jersey numbers, survey codes — is arithmetic without a subject.

The Expected Value Is a Number That Never Happens

Published 10/2/2026 · 4 min read · Everyday calculators

Antonio Delgado — Traditions & pastimes writer at OneKitly

Symbolism · Calendars

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In short

Expected value is each outcome multiplied by its probability and the products added: 10 × 0.5 + 20 × 0.3 + (−5) × 0.2 = 5 + 6 − 1 = 10. It looks reassuringly like one of the outcomes, and that is a coincidence — move the probabilities to 0.45, 0.35 and 0.20 and the answer becomes 10.5, which nothing that can happen will ever pay. The expected value is a weighted average of a set of possibilities, not a prediction of any one of them. Its meaning is entirely about repetition: it is where the average of many independent tries settles, and the number of tries needed for it to settle grows with how far apart the outcomes are. On a single try, the only honest statement is that one of 10, 20 and −5 will occur.

Outcomes of 10, 20 and −5 with probabilities 0.5, 0.3 and 0.2 give an expected value of 10. Shift the probabilities slightly and it becomes 10.5 — a result none of the three outcomes can produce.

The probabilities have to add up to one

A set that does not sum to one is not a probability distribution, and a calculation run on one is meaningless in a way that is easy to miss. Entering 50, 30 and 20 — percentages, which the wording did not forbid — used to return an expected value a hundred times too large, with nothing to indicate a problem. The tool now checks the sum and returns nothing rather than a wrong number, because normalising the input silently would have rescued the percentage case while quietly repairing a typo like 0.5, 0.3, 0.1 into a distribution nobody meant. If the answer comes back empty, the probabilities are the place to look.

A positive expected value is not a reason to bet the rent

Expected value treats every unit of money as worth the same as every other, which is a good approximation for small stakes and a poor one for large ones. The loss of a month's rent is not offset by an equal chance at a month's rent, because one of those outcomes removes a home and the other adds a windfall. That asymmetry is why a favourable average is a necessary condition for a good bet and never a sufficient one — the other half of the question is how many tries the average needs to assert itself, and whether the worst plausible run of luck along the way leaves anything to keep playing with.

Contribution

A single-number bet on a 37-pocket wheel
Outcome	Probability	Contribution
Win, +35 units	1 in 37	+0.946
Lose, −1 unit	36 in 37	−0.973
Expected value	—	−0.027

Worked with our own calculator

Expected value calculator

Given

Outcomes (comma-separated): 10, 20, -5
Probabilities, adding up to 1: 0.5, 0.3, 0.2

Result

Expected value: 10

These figures are produced by the calculator below, not typed in by hand — they are recomputed whenever the tool changes.

Run it on your own figures →

Frequently asked questions

How many tries before the average settles?: It depends on the spread of the outcomes rather than on their average, and it is usually far more than intuition suggests. The uncertainty around an observed average shrinks with the square root of the number of tries, so cutting it in half needs four times as many. For a bet whose outcomes are close together a few dozen tries give a stable picture; for one that pays thirty-five times its stake once in thirty-seven, thousands of spins can still end well away from the expected value in either direction, which is exactly why a run of good luck proves nothing.
Can outcomes be things other than money?: Yes — any quantity that can be added and multiplied works, which is what makes the idea travel so far. Expected delivery times, expected number of defects in a batch, expected repair costs across a fleet and expected calls per hour at a helpdesk are all the same calculation with different units. The one requirement is that adding the quantity has to mean something: expected temperature across three rooms is a legitimate average, while an expected value computed over categories with numeric labels — postcodes, jersey numbers, survey codes — is arithmetic without a subject.

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This article explains an arithmetic idea and uses a roulette wheel because its numbers are exact and public. It is not a guide to gambling and contains no system: the calculation below shows that every single-number bet loses 2.7 % of its stake on average, and no sequence of them changes that. If gambling has stopped being a pastime, the helplines that exist in every country are the right place to go, not a calculator.

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