Multiply, then add
The expected value is the sum of every outcome value multiplied by its probability. Nothing more. A fair six-sided die has an expected value of (1+2+3+4+5+6) divided by 6, which is 3.5 — a number the die can never actually show, which is the first useful thing to notice about expected values. They describe the long-run average of many attempts, not any attempt in particular.
The variance falls out of the same table: take the expectation of the squared values, subtract the square of the expected value, and the standard deviation is its square root. Where the expected value tells you where the average lands, the standard deviation tells you how far a typical single attempt sits from it. The two together are what a decision needs; either one alone misleads.
Enter values one way or the other, never both
There are two consistent conventions for a game with an entry cost, and mixing them double counts. Either put gross payoffs in the value column and fill in the cost field, or put net results in the value column — negative for the losing outcomes — and leave the cost field empty. For a one dollar ticket paying ten thousand with probability 0.001, the first way is 10000 and 0 with a cost of 1, and the second is 9999 and minus 1 with no cost. Both give a net expected value of 9. Do both at once — 9999 and minus 1 with a cost of 1 as well — and you get 8, having paid for the ticket twice.
What the numbers say about common gambles
| Wager | Return per unit staked | Why |
|---|---|---|
| European roulette, single number | about 0.973 | Pays 35 to 1 on a 1 in 37 chance |
| American roulette, single number | about 0.947 | Same payout, 38 pockets |
| Typical state lottery | about 0.50 | Around half the take is returned as prizes |
| Blackjack, basic strategy | about 0.995 | Player decisions cut the edge to well under a percent |
Every one of these is below 1, and no betting system changes that. Doubling after a loss, raising after a win, waiting for a number that is due — all of these rearrange which attempts carry which stakes, and the expected value of a sum is the sum of the expected values regardless of the arrangement. A system can reshape the distribution, usually trading many small wins for a rare catastrophic loss, but it cannot turn a negative expectation positive.
Variance is the part that ruins people
A positive expected value is not sufficient reason to take a bet. Consider a wager that doubles your entire net worth with probability 0.5 and takes all of it with probability 0.5. The expected value is a 50 percent gain, and taking it repeatedly leads to ruin with probability approaching one, because a single loss ends the sequence and no future gain recovers from zero. Expected value is an average over a scenario you may not survive long enough to experience.
This is why the size of a bet relative to your resources matters as much as its edge, and it is what the Kelly criterion formalises: stake a fraction of the bankroll proportional to the edge, so no single loss removes your ability to keep playing. It is also why the standard deviation sits next to the expected value here rather than tucked away. If the standard deviation is several times the expected value, then over any run you can actually afford, the expected value tells you very little.
Negative expected value can still be the right choice
Insurance has a negative expected value for the buyer by construction — the premium has to cover expected claims plus costs and margin, or there would be no insurer. People buy it anyway, and they are right to. The purpose is not to make money on average; it is to convert an unaffordable rare loss into an affordable certain one. That is variance reduction, and it is worth paying for whenever the bad outcome is one you could not absorb.
So the rule is not maximise expected value; it is maximise expected value subject to surviving the downside. Take the positive-expectation bets you can repeat and afford to lose. Pay to remove the negative-expectation risks that would end you. Decline the positive-expectation bets that could. Insurance sits in the middle of that list, and it is the one case where a negative number in the box above is not an argument against.
Questions people ask
Do the probabilities have to add up to 1?
For the expected value, only if the omitted outcomes have non-zero values. Entering a single line for a lottery win at probability 0.001 and value 10000 gives the correct expected value of 10, because the missing 99.9 percent is worth nothing and contributes nothing. The variance is a different matter: it is computed from the outcomes you listed, so leaving out the zero-value line changes the second moment. This page reports the sum either way so you can see what you left out, and warns when it exceeds 1, which is always a mistake rather than a shortcut.
How do I handle an outcome with a range of values rather than one number?
Use its own expected value in the value column, if all you care about is the overall average. A payout described as somewhere between 500 and 1500 with no reason to favour any part of that range has an expected value of 1000, and putting 1000 in the row gives the correct overall expectation. What that loses is variance: collapsing a range to its mean understates the true spread, so the standard deviation reported here will be too small. If the spread is the thing you are trying to assess, split the range into several rows with their own probabilities instead.
Does a big lottery jackpot rollover ever make the ticket worth buying?
The headline expected value can cross the ticket price when a jackpot rolls over far enough, but the realistic figure rarely does. Three things eat it: tax on the prize, the lump sum being substantially less than the advertised annuity total, and jackpot sharing — a large jackpot draws more tickets, which raises the chance a winning combination is picked by more than one person and the prize is split. That last effect scales with the same publicity that created the rollover. And even where the expectation does cross the line, the standard deviation is millions of times the mean, so no individual can play often enough for the average to mean anything.
Is expected value useful outside gambling?
It is the standard tool for any decision with known probabilities and quantifiable outcomes — whether to carry a spare part against the cost of downtime, whether a warranty is priced fairly, whether to over-order material against the cost of a second delivery. The honest limitation is that real decisions usually have probabilities you are guessing at, and an expected value computed from invented inputs inherits their error without displaying it. The discipline of writing the outcomes down and assigning numbers is often worth more than the figure it produces, because it makes the guesses visible and arguable.