Expected Value & the Long Run

Expected value (EV) is the average outcome of a wager, each possible result weighted by its probability. For every casino game with a house edge, EV is negative: on average, EV equals minus the house edge times the stake. The law of large numbers guarantees that as bets accumulate, the realised average converges on the EV — which is why, over time, the edge and not luck decides the result.

Multiply each possible outcome by its probability and add them up. If the total (net of stake) is negative, the bet loses money on average. Casino games are engineered so that this sum is always negative for the player — that negative number is the house edge in currency terms.

The law of large numbers says the realised average return approaches the EV as the number of bets grows. Short sessions swing around it (variance); long play grinds toward it. There is no ordering or sizing of negative-EV bets that turns their sum positive.

Martingale, D'Alembert and the rest rearrange when you win and when you lose, but they never change the per-bet EV. They trade a high chance of a small win for a small chance of a catastrophic loss — the sum of negative-EV bets is still negative EV, just with uglier variance.

Advantage play — card counting, certain full-pay video-poker tables, or matched betting on a promotion — can make EV positive. That is precisely why operators restrict or ban it. Absent a genuine edge, EV is negative by design, and that design is the business model.