Expected valuethe long-run average
Each outcome times its probability, added up: what average luck pays, and how casinos price everything.
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A carnival booth offers a deal: pay two dollars, roll one die, and win whatever number comes up, in dollars. Roll a six, collect six dollars. Roll a one, collect one dollar. Is that worth two dollars a try?
One roll is anyone's guess, but a thousand rolls are not. A fair die lands on each face about one sixth of the time, so six typical rolls pay out one, two, three, four, five, and six dollars — twenty-one dollars in total. Twenty-one over six rolls comes to three dollars fifty per roll.
That long-run number has a name: the expected value. Take each outcome, multiply it by its probability, and add up the pieces. The one contributes a small slice, the six a big one, and the slices stack up to exactly three dollars fifty.
Now judge the deal. The game hands back three fifty per roll on average, and a ticket costs two dollars — an average roll leaves you up a dollar fifty. No carnival would ever run that game. Raise the ticket to five dollars, and the same three fifty coming back leaves the player down a dollar fifty per roll. Any single roll can still win. The average cannot.
Expected value is what average luck pays. One roll can land anywhere, but pile the rolls up and the running average locks onto three fifty. That's how a casino prices every game: set the ticket so the expected value tilts the house's way, and let the long run collect. Players bet on single rolls. The house bets on the average — and the average always arrives.