Expected Value: The Only Number That Matters
The expected value (EV) of a bet is the weighted average outcome over all possible results, where each outcome is weighted by its probability. A positive EV bet gains money on average; a negative EV bet loses it. In casino gambling, every standard bet has negative expected value for the player — that is the definition of the house edge.
Roulette: Pure Probability in Action
Roulette is the clearest example of casino mathematics because the probabilities are entirely fixed and visible — the wheel has a known number of pockets. The house edge comes solely from the zero (and double-zero in American roulette), which causes all even-money bets to lose at a slightly higher rate than they win. *La Partage rule: on even-money bets, if the ball lands on zero, only half the bet is lost — halving the house edge on those specific bets. The key insight: the house edge is the same regardless of what number or colour you bet on, and regardless of what has happened in previous spins. Each spin is an independent event. The roulette wheel has no memory. The "gambler's fallacy" — believing that after a long run of red, black is "due" — is mathematically false. Previous results carry zero predictive power for future outcomes. If red has come up ten times in a row, the probability of red on the next spin is still 18/37 — exactly as before. The wheel has no memory. Betting on black because it is "due" is the gambler's fallacy, and it has cost fortunes throughout history. The casino is aware of this cognitive bias and actively encourages belief in it by displaying the recent history of outcomes on boards next to each table.
Blackjack: Where Skill Actually Reduces the Edge
Blackjack is unusual among casino games because the house edge depends significantly on how the player makes decisions. With perfect basic strategy — the mathematically optimal action for every hand versus every dealer upcard — the house edge drops to around 0.5% in favourable rule sets. With poor play (hitting 16 vs a dealer 2, for example), the edge can exceed 4%. Edward Thorp's 1966 book Beat the Dealer introduced the first mathematically rigorous card-counting system, demonstrating that because cards are dealt without replacement, the composition of the remaining deck changes the probabilities. A deck rich in tens and aces is slightly favourable for the player. Card counting is not illegal, but casinos are private establishments and may refuse service to anyone they suspect of counting. The house edge on blackjack for a skilled card counter is positive — but only in games with favourable rules, and only by fractions of a percent that require very large bankrolls and thousands of hours of play to realise.
Slot Machines: The Hidden Return to Player
Slot machines have no visible probabilities — the player cannot calculate the odds by inspection. The key metric is the Return to Player (RTP) percentage, which is the theoretical long-run fraction of all wagers returned to players as winnings. The house edge is simply (1 − RTP).
Variance, Volatility, and the Risk of Ruin
Expected value tells you what happens on average over infinite trials. In the short run, variance determines what actually happens. High variance means large swings around the mean — you might win big in a session, or lose your entire bankroll. The probability of losing your entire bankroll before reaching a profit target is calculable:
The Kelly Criterion: Optimal Betting for Positive EV Situations
The Kelly criterion is not a casino strategy — it is a mathematically optimal money management formula for situations where the player has a genuine positive expected value (sports betting with superior information, financial trading, card counting in favourable conditions). It maximises the long-run growth rate of a bankroll: