The Framing
Casino games are games of chance with a built-in house edge. No strategy, system or reading of past results 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.
This article looks at three linked ideas from probability: independence, the gambler’s fallacy and the law of large numbers. No data is used or invented; each point follows from the rules of the games.
Independent Events
Two events are independent when the outcome of one does not change the probability of the other. Successive spins of a roulette wheel are independent. So are successive rolls of dice and flips of a coin. The wheel has no memory of where the ball landed last time.
For a single-zero wheel, the chance of red on any spin is 18/37, about 48.65%. It remains 18/37 after one red, after five reds, and after twenty. The wheel does not know what came before.
Not all events in games are independent, however. Cards drawn from one deck without replacing them are dependent. If an ace has already been dealt, the chance of another ace is lower, because fewer remain. The difference between dependent events and independent events is why card games and wheel games call for different reasoning. Even in card games, though, a shuffled deck gives no hint about where it is heading unless the cards already seen are tracked, which is a different thing from a “streak”.
Calculating a Streak
Because spins are independent, the chance of a sequence is the product of the individual chances. The chance of five reds in a row on a single-zero wheel is (18/37)^5.
Stepwise: 18/37 is about 0.4865. Squared, about 0.2367. Cubed, about 0.1151. To the fourth, about 0.0560. To the fifth, about 0.0272. That is about 2.7%, or roughly 1 in 37.
A streak like that is uncommon enough to feel remarkable. Still, across many spins, such streaks will occur, and they should. In a long sequence of results, the absence of any streaks would be strange.
The Gambler’s Fallacy
The gambler’s fallacy is the belief that after a streak, the opposite result becomes more likely, as though the process were balancing itself. After five reds, the thought goes, black is “due”.
The error is easy to state. The chance of five reds in a row was about 2.7% before the sequence began. But after five reds have already happened, the question for the next spin is different: what is the chance of red on this spin? That is still 18/37. The earlier five spins are history, and they are not part of the next outcome.
Worth noticing: the fallacy is a mistake about independence. People apply the logic of dependent events (such as cards leaving a deck) to a process where it does not apply. A related mistake runs the opposite direction: expecting a streak to continue because it is “hot”. Both attribute a memory to a process that has none. The same reasoning is why a slot machine is never due to pay after a run of losses.
The Law of Large Numbers
The law of large numbers says that as the number of independent trials grows, the proportion of times an outcome occurs tends to approach its probability.
This is often misunderstood as a claim about “evening out”. A hypothetical makes the difference clear.
- After 10 flips of a fair coin, suppose there are 7 heads. The proportion is 70%, which is 20 percentage points above 50%. In terms of counts, heads are 2 more than tails would be at an even split of 5.
- After 1,000 flips, suppose there are 520 heads. The proportion is 52%, only 2 percentage points off. But in terms of counts, heads are 20 above the even split of 500, which is a larger absolute gap than before.
The proportion got closer to one half, even though the raw count difference grew. Nothing compensated for the earlier heads. The later flips simply carried on at 50% and diluted the early excess in proportion terms.
That is the correct reading: the law concerns averages and proportions, and says nothing about forcing totals back to zero. For a casino bet, it means the average result per bet moves toward the expected value, which is negative for the player. It does not mean a losing run will be offset by a winning one.
Why This Helps the House, Not the Player
A casino game is built around a small, steady advantage on every bet. The law of large numbers means that the more bets are placed, the more reliably that advantage shows up in the averages. A venue handling enormous numbers of bets sees results close to the theory. A single player placing a limited number of bets sees much more scatter, because small samples are noisy.
This asymmetry is what makes the edge reliable for the operator and unreliable, from one session to the next, for the player. It is also why a player’s good run is not evidence of a changed probability. Variation around the average is expected.
What Probability Can and Cannot Tell You
- It can say how likely each outcome is on a fair mechanism.
- It can say how often something will happen over many repetitions.
- It cannot say what will happen on the next spin.
- It cannot say whether a streak is about to end.
- It cannot be used to change the game’s expected return.
Reading the numbers correctly removes some false hopes, and that is a useful result in itself. The cards and the wheel do not owe anyone a turn of fortune, and the same is true of any random number generator.