Expected Value in One Sentence, Then a Warning
Expected value is a weighted average of outcomes. One sentence of definition, then the limits of the idea.
Expected value is the average result you would get if you could repeat the same uncertain choice many times, weighting each result by how likely it is. That is the whole definition. The rest of this article is about not stretching it into a promise. Poker conversations use the phrase as if it were a mood. It is a calculation, and only when the outcomes and the probabilities are actually known.
How those probabilities differ from odds against is in Odds and Probability Are Not the Same Number. Why a short run refuses to look like the average is in Variance in Poker. Nothing below tells you to put money in a pot. Adults who treat an average as a guarantee are the people the last section is for.
A coin, written out
Take a fair coin. Heads pays you one dollar. Tails costs you one dollar. The probability of each face is 1/2. The expected value is one half times one, plus one half times negative one. That sum is zero. Zero does not mean every toss lands in the middle. It means the average of a long series of tosses is pulled toward zero. Ten tosses can still finish ahead or behind. The definition did not fail. The sample was short.
A price that is worse than the chance
Suppose a fair six-sided die will either pay you a net gain of three dollars, if it shows a six, or cost you one dollar otherwise. The gain happens 1/6 of the time. The loss of one dollar happens 5/6 of the time. Expected value is 3/6 plus negative 5/6, which is negative 2/6, or about minus 33 cents per play, on average, over many plays. A single roll can still hit the six. The hit does not make the price fair. It makes you one of the cases the 1/6 already counted.
What the average does not pay you tonight
Nobody hands you the expected value in cash at the end of one trial. You receive one of the outcomes. In a card game the list of outcomes is much longer than heads or tails, and the probabilities are easy to misstate once other players' hidden cards matter. If you cannot write the outcomes and the probabilities, you cannot write the expected value. Saying a call is profitable without that list is a costume. It borrows the authority of arithmetic without doing any.
Even a correct figure is an average over repetitions you may not play, under conditions that may not hold. It is not a forecast of the session. It is not wages. It does not consider whether the stake was borrowed, whether you are tired, or whether the game is legal where you sit. Those facts can make a positive number a bad idea in real life. A negative number, clearly computed, is a reason to decline the bet. A positive number is not, by itself, a reason to accept it.
Use the sentence, then stop
Keep the definition and the two examples. Use them when someone treats a lucky hit as proof that a bad price was good. Then stop. Do not build a career story on a phrase. If the money matters, the prior question is whether you can lose the stake at all, which is a budget question, not an averaging question. Poker remains uncertain after the average is known. The average is a description. It is not an escort.
Frequently asked questions
Is expected value the same as the most likely result? No. In the die example the most likely result is losing the dollar. The expected value is a loss of about 33 cents, a number you will never receive on a single roll.
Can two people compute different values for the same pot? Yes, if they assume different probabilities. The phrase does not settle the argument. The count might.
Does a positive expected value mean I should play? No. It ignores your budget, the law, and the fact that you might not repeat the choice often enough to see the average. Declining is always available.