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. Understanding how events relate to each other explains the mechanics; it does not turn a game in the player’s favor.
The core distinction
Two events are independent if the result of one does not change the probability of the other. They are dependent if it does.
The difference matters because the way you combine probabilities depends on it. The sample space and basic multiplication rule from probability basics only work directly for independent events. For dependent events, you must update the probabilities as the situation changes.
Independent examples: dice, wheels, coins
A fair die has no memory. If it shows a six, the chance of a six on the next roll is still 1/6. A roulette ball likewise has no knowledge of earlier spins, so on a single-zero wheel the chance of red is 18/37 on every spin, whatever happened before.
For independent events, multiply.
- Two sixes in two rolls: 1/6 × 1/6 = 1/36, about 2.78%.
- Five reds in a row on a single-zero wheel: (18/37)^5, about 0.0272, or 2.7%.
That second figure is the chance of the five-spin sequence before any spin happens. Once four reds have already appeared, the chance of a fifth is just 18/37, about 48.65%. The wheel does not “correct” itself. Mistaking a rare sequence for a changed next-spin probability is the gamblers fallacy, which gets its own treatment in the roulette material.
Slot machines are designed to behave like this as well. A random number generator picks results with no memory of earlier spins, so a machine cannot be “due” to pay, a myth discussed under due machines. The independence of these results is the formal basis for the claim, in the broader piece on independent events, that past results do not change the odds.
A caution about real equipment
Independence is a property of an ideal device. A physical wheel with a flaw, or a poorly shuffled deck, could in principle depart from it, and this is partly why equipment is inspected and games are tested for fairness. For properly built and tested games, treating spins, rolls and software results as independent is the standard assumption.
Dependent examples: cards without replacement
A deck of cards does have memory, in the sense that a dealt card is gone. Each draw changes what remains.
Take two cards from a 52-card deck. What is the chance that both are aces?
- First card is an ace: 4/52.
- Given that, 3 aces remain among 51 cards: 3/51.
- Both: 4/52 × 3/51 = 12/2,652 = 1/221, about 0.45%.
Compare it with an independent version, where the first card is put back and the deck reshuffled before the second draw (replacement):
- 4/52 × 4/52 = 1/13 × 1/13 = 1/169, about 0.59%.
The numbers differ because, without replacement, the first ace removes one of the aces that could have appeared second. When the second probability is calculated given that the first event happened, it is called a conditional probability. Dependent events chain conditional probabilities together.
Another way to see it
Suppose a ten-value card appears first in a blackjack hand. The deck now has one fewer ten-value card, so the chance of another is slightly lower than before, and the chance of every other rank is very slightly higher. That is dependence in action: a single known card shifts the remaining deck.
Why dependence matters in card games
Because the composition of a deck changes as cards are used, later cards depend on earlier ones. This is the principle behind card counting, which tracks what has left a shoe to estimate how the remainder differs from a fresh deck. Whether this helps depends on the rules, the number of decks, how many cards are dealt before reshuffling, and many other conditions. Casinos may refuse service to players they suspect of counting, and rules vary by place. The mathematical idea is real, but it does not make a typical player a favorite, and this article gives no advice on whether to try it.
More decks dilute the effect. With a single deck, one removed card changes the odds more than it does in a shoe of several decks, which is one reason the number of decks is a key rule variation. When a deck is reshuffled after every hand, dependence across hands disappears, because each hand begins from a full deck.
Hands and bets versus cards
It helps to keep two layers separate.
- Within a hand, cards drawn from a deck are dependent: the later ones are affected by the earlier ones.
- Between hands, if the deck is freshly shuffled, the hands are independent of one another.
The same distinction applies to a roulette session. A single spin’s outcome is independent of the previous one, but the combined results of several spins follow the multiplication rule above.
A quick test
To decide which type you are dealing with, ask: “If I learned how the first event turned out, would I change my estimate for the second?”
- If no (a die roll, a spin, a software-driven reel), the events are independent.
- If yes (a card drawn from the same deck without replacement), they are dependent.
Knowing which rule applies keeps you from two typical errors: treating cards as if they were dice, and treating dice as if they had memory. Neither understanding changes the built-in edge of house-banked games, and results in any given session remain uncertain.