Gambling math: odds, house edge and probability
Gambling math reduces every casino game to four numbers: probability, payout, house edge and variance. This guide works through each with examples.
Probability basics
Gambling math starts with probability. A probability is favourable outcomes divided by all possible outcomes under the rules being used. With two dice, a total of 7 can be made in 6 of 36 ways, so the probability is 6/36 = 16.67%.
The key step is counting outcomes correctly, not guessing from how common a result feels. Many gambling mistakes start with the wrong denominator, such as counting only the outcomes the player can picture instead of every outcome the rules allow.
Independent events do not change each other. If a die roll has already happened, it does not alter the next die. For example, the probability of rolling a 6 on one die and a 6 on another is found by multiplying the chances, giving 1/36 = 2.78%.
Multiplication is used because both events must happen. This is why repeated attempts do not create a debt that the game must repay. A previous losing result can affect a bankroll, but it does not make the next random result more generous.
Dependent events change the remaining pool of outcomes. Drawing an ace or a king from a full deck uses addition because the winning groups are separate: 8/52 = 15.38%. Drawing two aces in a row from one deck without replacement is different because the first draw changes the second draw.
The probability is 1/221 = 0.45%. Card games, shoe games and any game where an item is removed from the pool need this kind of careful counting. The same idea explains why card combinatorics matters in blackjack and poker-style games, while dice and roulette outcomes reset after each completed round.
Odds and probability
Odds are prices or ratios that express a probability. The odds to probability formula depends on the format: decimal odds use the reciprocal of the price, fractional odds compare profit with stake, and American odds need conversion based on whether the price is positive or negative.
The table below shows the core relationships. The useful habit is to convert every price into an implied probability before comparing it with the real chance of the event.
Format | Example | Implied probability |
|---|---|---|
Decimal | 2.00, equal to fractional 1/1 | 50.00% |
Fractional | 5/1 | 16.67% |
Decimal | 4.00 | 25.00% |
American positive | +300, equal to decimal 4.00 | 25.00% |
American negative | -200 | 66.67% |
True odds describe the real chance of an outcome. Payout odds describe what the operator pays if that outcome wins. In European roulette, a straight-up bet wins with probability 2.70%, while the true odds against are 36:1.
The casino payout is lower than the true odds would require, and that gap creates the house edge of 2.70%. The same comparison applies to sportsbook prices, table games and slot paytables. If the payout is worse than the true probability would justify, the wager has negative expected value for the player.
How the house edge is calculated
House edge is calculated by multiplying each possible result by its probability and payout, adding the results, then comparing the average result with the stake. On a $10 straight-up bet on European roulette, the expected value is -$0.27. The same method applies to any bet where the probabilities and payouts are known.
The calculation is not a prediction for the next result. It is the average price of making the same type of wager under the same rules. A $10 red bet on American roulette has an expected value of -$0.53.
A $10 bet on a coin flip paying 1:1 has an expected value of $0.00. A $1 stake on a slot with 96% RTP has an expected value of -$0.04, and $1,000 total wagered at 96% RTP has an expected loss of $40.
These examples show the same structure: probability and payout create an average, even though any single round can finish above or below it.
This is the main reason betting systems cannot change the underlying maths. Changing stake size changes cash-flow risk, session length and the size of wins and losses. It does not change the probability of the next spin, hand or roll. If the game pays less than fair odds, the expected value remains negative before any external costs or restrictions are considered.
House edge and RTP
RTP shows the same relationship from the player side. A 2.70% house edge is equivalent to 97.30% RTP, and a 5.26% house edge is equivalent to 94.74% RTP. If a game has 96.5% RTP, the house edge is 3.5%.
The RTP guide explains why this is an average over betting volume, not a prediction for any single spin, hand or round. RTP is useful for comparing rules, but it says less about how volatile the path will feel.
House edge and casino hold
Casino hold is what a table keeps from chips bought or money dropped during play. It differs from house edge because players can buy in, win, re-bet, lose, buy in again, or cash out. The accounting result can be above or below the mathematical edge in a short period.
House edge is cleaner for comparing games because it is tied to the rules of each bet. Hold is useful for venue accounting, but it should not be used as a shortcut for the price of a single wager.
Variance and standard deviation
Variance explains why short-run gambling results can look nothing like the average. A negative expected value can still produce winning sessions, and a low house edge can still produce losing streaks. The house edge describes the long-run average cost of action.
Variance describes how widely results scatter around that average. They see a low edge and expect a calm ride, but a low average cost can still be attached to a wide spread of possible outcomes.
Standard deviation is one way to measure that scatter. A European roulette even-money bet has a standard deviation of 1.000 per unit bet. A European roulette straight-up bet has a standard deviation of 5.84 per unit bet.
The straight-up bet is much more volatile because most outcomes lose, while a rare winning outcome pays much more. Both bets have a negative average under European roulette rules, but they feel different because one produces frequent small changes and the other produces long losing stretches interrupted by occasional large wins.
After 100 European even-money bets of 1 unit, the expected result is -2.7 units, while the range covering about 95% of results is -22.3 to +16.9 units. For 100 European straight-up bets, the range covering about 95% of results is -117 to +112 units.
Both ranges include positive outcomes, but the average remains negative. The wide range also explains why short sessions are poor evidence about whether a game is fair. A lucky result is not proof of a beatable game, and an unlucky result is not proof that a fair random game has stopped being random.
Variance also changes how limits feel. A small stake on a high-variance wager may last longer than a large stake on the same wager, but the underlying edge per unit remains the same.
A player who wants slower swings usually needs smaller stakes relative to the bankroll and a lower-volatility bet type. That choice can reduce the chance of a fast loss, but it does not turn a negative expectation into a positive one.
Risk of ruin
Risk of ruin asks a different question from house edge. It asks how likely a bankroll is to hit the losing boundary before reaching a target. With a 50-unit bankroll, making 1-unit European even-money bets gives a 93.72% probability of ruin before doubling the bankroll. With 5-unit bets, a bankroll of 10 bets, the probability is 63.20% under the same double-or-bust target.
This does not make the larger stake a better bet. It changes the race between the losing boundary and the target. Larger bets give less room for adverse variance, make limits matter more, and can end a session quickly.
Risk calculations only make sense when the bankroll, stake size and goal are all defined. A target that sounds simple in words can still be mathematically hostile once the house edge and variance are included.
Risk of ruin is also a better tool than chasing rules of thumb. A staking plan may feel safer because it produces many small wins, or more exciting because it reaches a target faster when early results go well.
The missing question is always the same: what happens when the losing run arrives before the target? If the answer is a depleted bankroll, the staking plan has not removed the risk. It has only shaped how the risk appears during play.
Combinations in card games
Card games use combinations because the order, deck size and replacement rules change the chance of each hand. In a six-deck blackjack shoe, the probability of being dealt a blackjack is 4.75%. That figure comes from counting the card ranks that form blackjack against all possible starting hands under those shoe rules.
Blackjack also shows why rules matter. The same broad game name can cover different rule sets, and a small rule difference can change the value of a starting hand or the cost of a mistake.
Poker-style examples show the same idea. In a five-card deal, there are 2,598,960 possible five-card hands from 52 cards. The probability of a royal flush is 1 in 649,740.
Combinatorics matters because rare hands often have large payouts, but the payout must still be compared with the true probability. A large advertised prize is not enough. The right question is whether the prize is large enough for how rarely the hand appears.
Replacement is the key distinction for many players. A roulette spin does not remove a number from the wheel for the next spin. A card drawn from a deck without replacement changes the next draw because the composition has changed.
That is why a card example such as drawing two aces in a row is not solved the same way as rolling two dice. In one case the pool changes, in the other it resets or remains independent.
Bookmaker margin
Sportsbooks add margin by setting prices whose implied probabilities add to more than the available outcomes. In a football 1X2 market, the home win, draw and away win cover the possible match results. This example uses decimal odds. Converting each price into an implied probability lets the margin become visible, even before anyone discusses team strength, form or injuries.
Outcome | Decimal odds | Implied probability |
|---|---|---|
Home | 2.10 | 47.62% |
Draw | 3.40 | 29.41% |
Away | 3.60 | 27.78% |
The implied probabilities sum to 104.81%, called the book percentage or overround. Converting that book percentage with 1 - 1/book gives a bookmaker margin of 4.59%. The margin is built into the prices before any opinion about the match is considered. This is why quoted odds are not the same thing as a neutral probability forecast.
The same habit helps with promotions and boosted prices. A bigger price can still be poor value if the implied probability remains above the event's realistic chance. A shorter price can still be reasonable if the true probability is higher than the price implies. The hard part is estimating the true probability well enough. The conversion itself is only arithmetic.
Where to go next
This page is the map for the maths guides. House edge explained gives the edge for every common bet, odds and probability basics covers the formats in more depth and casino RNG shows how random results are produced.
Expected value in gambling works through money examples, casino games with the best odds ranks the games by house edge, betting systems tests staking claims and the gambler's fallacy explains why streaks do not make a result due. For the average cost of play from the player's side, see what RTP means.
Probability tells you how often outcomes occur under the rules, payouts tell you what the operator gives back, and house edge turns the two into an average price. Variance and risk of ruin explain why the route to that average can still be rough. Use the maths to price the entertainment, then set deposit and loss limits before you play.
Frequently asked questions
What is the difference between odds and probability?
Probability is the chance that an outcome occurs. Odds express that chance as a price or ratio. In gambling, payout odds can be worse than true odds, and that difference creates the operator margin.
How is house edge calculated?
List each possible result, multiply it by its probability and payout, then add the outcomes to get expected value. For a $10 straight-up European roulette bet, the expected value is -$0.27, which corresponds to a 2.70% house edge.
What does RTP mean?
RTP means return to player. It is the long-run average returned to players from money wagered. A 96.5% RTP is equivalent to a 3.5% house edge, but it does not predict the result of a single session.
Why do I lose even on low house edge games?
Low house edge reduces the average cost of play, but it does not remove variance. In 100 European even-money bets of 1 unit, the expected result is -2.7 units, while the range covering about 95% of results is -22.3 to +16.9 units. Winning sessions can happen, but repeated play still has negative expected value.