Casino games are games of chance with a built-in house edge. No strategy, system or timing changes the odds in the player’s favor over time, and most players lose money over time. Nobody should gamble money they cannot afford to lose. Break-even math describes what a payout would require; it does not mean any real game reaches it.
Two numbers, one question
Every bet has two basic features:
- Hit rate: how often the bet wins, as a probability or percentage.
- Payout ratio: how much is paid for each unit staked when it wins, such as 1 to 1 or 35 to 1.
A third idea ties them together. The break-even probability is the win chance at which the average result per bet is exactly zero. If the true hit rate is below that figure, the bet loses on average. If it is above, the bet wins on average. Casino bets are priced so that the true hit rate is below break-even.
Deriving the formula
Say a bet pays a to b: you win a units for every b units staked. Let p be the probability of winning. Over many bets, break-even means average gain equals average loss:
p × a = (1 − p) × b
Solving for p gives:
break-even probability = b ÷ (a + b)
For the common “X to 1” payout, b = 1 and the formula simplifies to 1 ÷ (X + 1). This is the same reasoning behind expected value, set to zero.
Worked examples
| Payout | Break-even probability |
|---|---|
| 1 to 1 | 1 ÷ 2 = 50% |
| 2 to 1 | 1 ÷ 3, about 33.33% |
| 5 to 1 | 1 ÷ 6, about 16.67% |
| 8 to 1 | 1 ÷ 9, about 11.11% |
| 35 to 1 | 1 ÷ 36, about 2.78% |
| 3 to 2 | 2 ÷ 5 = 40% |
Roulette straight-up
A straight-up bet pays 35 to 1, so break-even is 1/36, about 2.78%. On a single-zero wheel the real chance is 1/37, about 2.70%. On a double-zero wheel it is 1/38, about 2.63%. Both are below break-even, and the shortfall is the edge. The gap is small in percentage points but is paid for on every spin.
Even-money bets
A bet that pays 1 to 1 needs a 50% win chance to break even. Red on a single-zero wheel wins 18 times out of 37, about 48.65%, so it falls short by about 1.35 percentage points. The bet is “even money” in payout but not in probability, which is the theme of the piece on even-money bets.
Blackjack natural at 3 to 2
If a hand pays 3 to 2, break-even is 40%. This does not mean blackjack is a break-even game; the figure applies only to a single bet taken in isolation with a clean win-or-lose outcome. Real blackjack has pushes, doubles and splits. How ties are handled changes the picture, as discussed in push rules.
Hit rate alone is misleading
A high hit rate looks reassuring, but it says little by itself. Consider a hypothetical game:
- It pays 1 unit profit on a win, and loses 1 unit otherwise.
- The hit rate is 49%.
Break-even is 50%, so the bet loses 0.49 − 0.51 = −0.02 units per unit staked on average. It wins almost half the time and still loses.
Now consider a different hypothetical game where a win returns 5 units in total (4 profit plus the stake) and the hit rate is 19%. The return per unit staked is 0.19 × 5 = 0.95, so the average loss is 5% of the stake. Roughly one bet in five wins, and the bet still costs money on average.
The slot-style view
In a slot machine, many results return part of the stake, and a small share return larger sums. The relevant numbers are hit frequency (how often any win occurs) and the average size of the wins. A game could hit about 1 in 4 spins, which is 25%. To break even, the average total return on a hit would need to be 4 times the stake, since 0.25 × 4 = 1. Because small “wins” often return less than the stake, a high hit frequency does not guarantee that the stake is recovered. Details are in the article on hit frequency versus payout size, and the broader picture is in slot machine math.
A related point: a win that returns less than the stake still gets celebrated by the machine as a win. Counting such results as wins inflates the sense of success, and it is one reason players overestimate how they are doing.
Pushes and ties
When a bet can tie and return the stake, the break-even condition changes slightly. A simple way to handle it is to compare the chance of winning with the chance of losing, ignoring ties, if the payout is 1 to 1. With a payout of 1 to 1, the bet breaks even when the chance of winning equals the chance of losing.
Using break-even in practice
The formula gives you a quick test of any quoted bet:
- Convert the payout into its break-even probability.
- Find or calculate the true win probability.
- If the true probability is lower, the bet has a negative average result. The size of the gap, scaled by the payout, becomes the house edge.
This test confirms what the casino model already tells you: the house sets payouts a little below true odds. Even a bet that wins often, such as one that hits nearly half the time, can lose steadily. Outcomes in the short run remain uncertain, but the break-even comparison explains the direction of the long run.