Every wager has a true worth: multiply each outcome by its chance, add them up, and you get the EXPECTED VALUE — what you’d win or lose per bet on average, forever. A fair coin flip for a dollar is worth zero; a casino game is worth slightly LESS than zero, by design. It’s the number under every game at the table. Slide the odds and watch EV cross the line.
Expected value E[X] = Σ pᵢ·xᵢ is the probability-weighted mean of the outcomes — the long-run average per trial by the law of large numbers. For a +1/−1 bet, EV = p·(1) + (1−p)·(−1) = 2p − 1: positive above p=0.5, zero at a fair game, negative below. American roulette on red is (18/38)(1)+(20/38)(−1) = −5.26%. A fail-loud self-check throws unless a fair 50/50 gives EV=0 and roulette-red gives −5.26%. ◆ real probability, node-verified.
EV is the long-run mean; a single session swings around it (that variance is why gambling feels winnable). A positive EV doesn’t prevent short-run ruin — see [[the-gamblers-ruin]] and [[the-kelly-criterion]].