Independence
Events are independent when learning one result does not change the probability distribution of another. Proper roulette spins and RNG slot outcomes are modeled this way.
The chance of ten reds in advance is different from the chance of red on the next spin after nine reds have already occurred.
The fallacy
The gambler’s fallacy expects short samples to self-correct immediately. After a run of Banker results, Player is not owed a balancing win.
Increasing a wager because an outcome is due changes exposure without changing the underlying probability.
Finite decks
Card games are not perfectly independent because exposed cards change the remaining composition. That does not validate visual streak systems.
A defensible conditional calculation must identify the removed cards and recompute the distribution.
Random does not mean alternating
Random sequences naturally contain clusters and streaks. A perfectly alternating record would itself be suspiciously structured.
Evaluate a system against probability and payoff rather than whether its story sounds balanced.
Working checklist
- Ask whether trials are independent
- Separate completed sequence odds from next-event odds
- Use actual card removal when relevant
- Ignore due language
- Do not raise bets to force balance
This lesson explains general gambling mathematics and uses rounded examples. A game-specific figure applies only to the rules, pay table, strategy, and denominator stated with it. Last reviewed September 20, 2026.