The short answer
Under the title of this article, the honest summary is that probability explains casino games very well and cannot be used to win at them in the long run. The reason is a single idea: on every standard bet, the average result for the player is negative.
Expected value in one formula
The expected value (EV) of a bet is the average result per bet if you could repeat it many times. For a bet that wins an amount W with probability p and loses a stake S with probability 1 - p:
EV = p × W - (1 - p) × S
A bet is fair when EV is zero. It has negative expected value when EV is below zero, which describes every standard casino wager. It is positive only where the odds favour the player, and in the casino’s standard games the rules are set so that they do not.
A fair bet and a shortchanged one
Illustration with a die. Suppose you stake 1 unit that a fair six-sided die will show a 6.
- If the bet pays 5 to 1 (you win 5 units): EV = (1/6) × 5 - (5/6) × 1 = 5/6 - 5/6 = 0. That is a fair bet.
- If the bet pays only 4 to 1: EV = (1/6) × 4 - (5/6) × 1 = 4/6 - 5/6 = -1/6, about -0.167 per unit staked.
The second bet looks like the first with a payout slightly lower. It turns an average of zero into a loss of about 16.7% of the stake. Casino bets are priced like the second one. Roulette follows the same pattern. A 35 to 1 payout on a 1 in 37 chance gives -1/37, about -2.70%, which is the house edge on that bet, and the same principle applies whatever the game.
Even-money bets
Take an even-money bet that wins with probability 0.48 and loses with probability 0.52, as a simple illustration (not any specific game). Per unit staked:
EV = 0.48 × 1 - 0.52 × 1 = -0.04
On average, the player loses 0.04 units per unit staked, or 4%. Over 100 bets of 10 units, the total staked is 1,000 units, so the expected loss is 4% of 1,000 = 40 units.
Two things follow from this:
- Expected loss grows in proportion to the total amount wagered. Betting more often, or with larger stakes, increases it.
- Because it is proportional, scaling the stake does not change the sign. A negative number multiplied by any positive amount is still negative.
Why “using probability” does not flip the sign
Common ideas for turning the maths around, and why they fail:
- Choosing the best bet. Knowing which bets have a lower edge is useful: it reduces how fast you lose on average. But the best of a set of negative bets is still negative.
- Combining bets. Placing several bets on the same round adds their expected values. A sum of negative numbers is negative. Covering many outcomes does not remove the edge, it spreads it.
- Waiting for a pattern. Outcomes of independent events such as roulette spins or dice rolls do not follow patterns that you can wait for. The belief that a long run of one result makes the opposite “due” is the gambler’s fallacy.
- Changing stake sizes. A scheme for varying stakes changes the pattern of wins and losses, but not the average per unit staked. Staking plans of this kind are the betting systems people often try, and they leave the average unchanged.
- Staking more. More staked means a larger expected loss, as shown above.
For a game of pure chance with a fixed payout table, EV per unit staked is a property of the game, so the player’s choices cannot alter it, beyond which bet to take.
Where skill changes the picture
Some games involve decisions, and there the situation is more nuanced. In blackjack, for instance, the choices a player makes affect the average result, and the house edge depends on the rules and on how closely play follows the best choices. Card counting is a further special case with its own limits. In these games, good decisions reduce the average loss; for the typical player it still remains negative.
In games against other players, such as poker, you are not up against a fixed payout table, so the sign of the average depends on your skill relative to the others and on fees taken by the venue. Even then, results over short periods are dominated by chance.
Short-term results versus the average
The negative EV is an average. In a single session a player can finish ahead, and many do. Over a small number of bets, the variability (measured by the standard deviation) is much larger than the expected loss, so winning is quite possible. That is why casino games feel winnable. The edge shows up over a large number of bets, and the more you play, the more closely your total result is likely to track the average: it is the law of large numbers working against the player.
A short winning streak is therefore not evidence that a method works, and it does not mean the next bet will be better or worse.
A realistic use of probability
If probability cannot create a winning method, what is it for?
- Understanding the real cost of play, as an expected loss per unit staked.
- Comparing games and bets on their edge before deciding whether to play at all.
- Spotting misleading claims about systems, patterns or guaranteed results.
- Setting spending limits that treat the money as the cost of entertainment, and sticking to them.
Gambling can be harmful, and outcomes are always uncertain. Never bet more than you can afford to lose, never borrow to play and avoid chasing losses. If gambling is causing problems for you or someone you know, contact a local support service.