Casino probability basics

Casino probability basics: counting the outcomes

Almost every probability in a casino comes from counting. A roulette wheel has a fixed number of pockets, a pair of dice has a fixed number of combinations and a shoe has a known number of cards. If every outcome is equally likely, the chance of any event is the number of outcomes that produce it divided by the total. No advanced maths is needed, just care in counting.

1. Define the event exactly, for example 'the ball lands on a red number'.
2. Count all equally likely outcomes: 37 pockets on a single-zero wheel.
3. Count the outcomes that give your event: 18 red pockets.
4. Divide: 18 ÷ 37 = 0.4865, or 48.65%.
5. Compare with the payout: red pays 1 to 1, but a fair price for 18 in 37 would be slightly more. That gap is the house edge.

Every page in our casino maths section builds on this step, from expected value to variance.

Writing probability as fractions, percentages and odds

The same chance can be written in several ways, and casinos tend to use the payout format rather than the probability format, which hides the gap between the two. Learning to switch between them makes any table easier to judge. Our guide to odds formats covers fractional and decimal odds in more depth.

In every row, the casino's payout is a little shorter than the true odds. That is the whole business model, and our page on true odds against payout odds works through more examples.

Combining chances: the 'or' rule and the 'and' rule

Two simple rules cover most casino questions. When events cannot happen together, the chance that one or the other happens is the sum of their chances. Landing on 7 or 8 at a single-zero wheel is 1/37 + 1/37 = 2/37, which is why a split bet covers 2 numbers and wins 5.41% of the time.

When events are independent, the chance that both happen is the product of their chances. Casino Math defines independence plainly: events are independent if the occurrence of one has no effect on the probability of the others. Two reds in a row on a single-zero wheel is 18/37 × 18/37, about 23.7%. Five in a row is about 2.72%, and ten in a row about 0.074%.

The 'at least once' question uses a third trick: work out the chance of the event never happening, then subtract from 1. The chance that a chosen number fails to land in one spin is 36/37. Across 37 spins it fails every time with probability (36/37)³⁷, about 36.3%, so it lands at least once with probability about 63.7%. Many people expect a number to appear once in every 37 spins, but more than a third of the time it does not.

The eight-losses figure matters for anyone tempted by the Martingale: a run that wipes out a doubling sequence is rare on any single attempt, about 1 in 207, but over hundreds of sequences it becomes likely to happen at some point.

Dice probability for craps and sic bo

Two dice have 36 equally likely combinations, because each of the six faces on the first die can pair with each of the six on the second. Totals in the middle can be made in more ways, so they come up more often. Six combinations make 7, which is why 7 has a 1 in 6 chance on any roll and sits at the centre of craps.

Three dice, as used in sic bo, have 6 × 6 × 6 = 216 combinations. Six of those are triples, so any triple has a 6 in 216 chance, or 2.78%. Small and Big bets lose to a triple, which leaves 105 winning combinations out of 216 for each, a 48.61% chance. Our dice probability page extends this to every total.

Card probability: when the past does matter

Cards are the one place where earlier results change later chances, because a card dealt from a shoe is not put back until the shuffle. In a single 52-card deck, the chance that the first card is an ace is 4/52, about 7.69%. The chance of a two-card blackjack from a fresh single deck is 2 × (4/52) × (16/51), about 4.83%: an ace then a ten-value card, or the reverse.

Once some cards are gone, those figures shift slightly. That is the theory behind card counting, which our card counting page covers as history and theory only. In practice, online RNG blackjack reshuffles for every hand, so each deal starts from a full shoe, and live tables generally use shuffling methods that defeat it. See continuous shuffling and the blackjack odds page for the numbers.

Independence, the gambler's fallacy and the hot hand

Wizard of Odds puts it bluntly in its betting systems analysis (opens in a new tab): every spin in roulette and every toss in craps is independent of all past events. A wheel has no memory, so after ten reds the chance of red on the next spin is still exactly 18/37.

Two well-studied mistakes push people the other way. The gambler's fallacy is the belief that an outcome which has not happened for a while is overdue. Economists Rabin and Vayanos, in their paper on the gambler's and hot-hand fallacies (opens in a new tab), describe it as expecting random sequences to reverse, such as expecting tails after a run of heads. The hot-hand fallacy is the opposite: believing a streak will continue, as many basketball fans believe about a player who has made several shots in a row.

In a casino, both fallacies lead to the same error: thinking past results tell you something about the next one. Electronic 'hot and cold numbers' displays at roulette tables record history, nothing more. Our page on hot and cold numbers explains why they carry no predictive value.

In the 37-spin example above, a number missing for a whole cycle is completely normal: it happens about 36% of the time. A 'missing' number is not a 'due' number.

UK rules on random outcomes

For online games in Great Britain, the Gambling Commission's RTS 7 standard on random outcomes (opens in a new tab) requires results to be acceptably random and forbids adaptive behaviour, meaning a compensated game that changes its probabilities to make up for past results. So an online slot or RNG roulette game cannot 'tighten' after a big win or 'loosen' after a cold spell. The standard also says a simulated device, such as a virtual wheel, should have the same theoretical probabilities as the real one.

That rule covers remote games. Some older land-based fruit machines in the UK have historically used compensated payout models, so the rule should not be read as applying to every machine ever made. Our pages on random number generators and do online slots have memory go further.

Probability in the short run and the long run

Probability describes proportions over many trials. Casino Math states the law of large numbers as the actual proportion of occurrences eventually approaching the theoretical probability, and illustrates it with coin tosses: the chance of 60% or more heads is 0.377 with 10 tosses, 0.028 with 100 and vanishingly small with 1,000. Short runs wander widely; long runs settle. Crucially, settling happens by dilution, not by correction, so a shortfall of reds is never 'paid back'. See our law of large numbers page.

Probability myths at the casino

• 'After five blacks, red is more likely.' The chance is 18/37 on every spin, whatever came before.
• 'Betting several numbers improves my odds overall.' It raises the chance of winning a spin but lowers the payout in proportion, so the edge is unchanged.
• 'Slots pay out at certain times of day.' UK online games may not adjust their odds, so timing makes no difference. Our answer on slot payout timing covers this.
• 'A rare result means the game is rigged.' Rare events happen all the time across millions of rounds. Ten reds in a row is about 1 in 1,347 on any given run of ten spins.

Probability and the casino offers here

The partner offers on this page are mainly for slots, and none changes the probability of any outcome. Casumo, 18+: opt in for a 50% bonus up to £100 on a first deposit of £20 or more, wagered 10x within 7 days, with a £5 maximum bet and game, play and payment restrictions. 10bet, 18+: a 100% match up to £25 on a first deposit within 7 days of sign-up, 10x wagering on slots within 3 days, bonus winnings capped at £1,000, Skrill and Neteller excluded. Coral, 18+: deposit and bet £10 on slots to get 100 free spins at 10p on selected games, accepted within 48 hours and valid for 7 days, with no wagering on winnings and some payment methods excluded.

Terms apply to every offer. Table and live games are often excluded or count less towards wagering, so check before using a bonus on roulette or blackjack.

18+. Every casino game has a built-in house edge, so over time you should expect to lose what you stake. Set a budget before you play and stop when it’s gone: safer gambling tools and support. Free, confidential help: National Gambling Helpline 0808 8020 133.