How does the casino get its edge on the bets you make?

Most folks you might ask, civilians you encounter in the real world as well as solid citizens you meet in the casino, think the bosses get their edge by winning more bets than they lose. Astute players can readily identify propositions they have a greater chance to win than lose, however. For instance, at single-zero roulette, pretend someone splits a bet evenly between two 12-number columns. The probability is to get 24 wins for every 13 losses. The odds, per se, accordingly favor winning by 24-to-13. In a "fair" – no edge – game, $24 total wagered this way would be paid $13. The actual payoff is only $12 with the disappearing dollar accounting for the edge.

House advantage ultimately boils down to the offset between payoff on the amount that can be lost and the odds against winning. Consider Red at single-zero roulette, on which "the house has more chance of winning than losing" rings true. This bet succeeds on any of the 18 red and fails on any of the 18 black numbers or the one green zero, so the probabilities are 18/37 (48.6 percent) for joy and 19/37 (51.4 percent) for sorrow. Players are, indeed, more likely to lose than win. But the essential factor is that in a fair game, for every dollar bet they'd win (19/18) x $1 or just under $1.06. Instead, they get $1. The missing $0.06 is what the house earns on edge.

A $5 Place bets at craps illustrates how the house winning more often than it loses isn't the sole determinant of edge. The odds are six losses for every four wins , 6-to-4 against the player. The bet would be fair were the payoff for a $5 wager (6/4) x $5 or $7.50. It isn't. The house pays $7.00. Edge is in the shortchanged $0.50, not the 6-to-4 chance against winning in and of itself. Slots likewise derive edge from the payoff-odds offset, although the values have to be reckoned for all possible outcomes. Say a hypothetical machine has the payoffs and probabilities shown in the accompanying table for $1 bet. Expectation at each level is the payoff times the probability. The sum of these expectations is the average fraction of the total wagered the casino withholds. For this situation it's 5 cents per dollar bet. Were players to lose on 57.5 percent of all pulls and win a buck on 40.63 percent, total expectation would be zero and the machine would be fair.

Source of house advantage for a hypothetical three-level slot machine

payoff	probability	expectation
$100	  0.01%	          $0.01
10	  1.56%	           0.16
1	  38.43%	   0.38
-1	  60.00%	  -0.60
overall	  100.00%	 -$0.05

Blackjack offers an alternate view of the same effect. Players exceeding 21 lose at once, no matter what dealers do. So dealers win double-busts. Under dealers' rules, busts have 28 percent probability. Were players to mimic these rules, the chance of double busts would be 28 percent of 28 percent – 7.8 percent. Win-loss probabilities would be 46.1 versus 53.9 percent, with a fair game paying (.539/.461) x $1 or $1.17 per dollar and not even money. Payoffs of 3-to-2 for uncontested blackjacks, pair splitting, doubling, and standing below 17 shave the $0.17 deficit.

Occasionally, house edge is achieved with a commission on winning bets. In baccarat, for example, a bet on Banker is slightly more apt to win than lose. With the nominal even-money payoff, players would have an edge. However, the dealer takes 5 percent commission, such that the actual amount paid is $0.95 on the dollar. This shifts the advantage back to the casino. Edge in games where players look at their hands, then decide whether to fold or pony-up for more, also follow the payoff-odds paradigm although the principle may not be immediately apparent. Folding loses the primary bet. The second wager moves players to the next phase where they compare hands against the dealer's. Four results can occur. 1) The hands are equal – a push, 2) the dealer's hand beats the player's – both wagers lose, 3) the player's hand beats the dealer's – both wagers win, 4) the dealer's rank is below the minimum needed to "qualify" – the primary wager wins and the secondary wager pushes. The key is that when dealers don't qualify, their hands are so weak that they're more likely to lose than win. Absent this provision, the chance would increase of players winning two rather than one unit and the game would become fair.

Every once in a while, a casino bean counter gets a brainstorm about collecting fees from players at the outset every round of a game. This "juice" applies whether the player eventually wins or loses the hand so, when averaged out, it's higher than the same percentage deducted from winning bets. The sum involved may seem small. But at low-limit tables, it can represent a high fraction of the wager and accordingly a usurious edge. If you're playing $2 blackjack, a $0.10 fee is 5 percent on top of whatever advantage the house would otherwise have on the action. These tables are best avoided. Fortunately, for the naive, when schemes like these are tried, they're typically short-lived. Word does get around. And players needn't be experts to notice how quickly their bankrolls dwindle in these games then stay away in droves, adhering to the exhortation exquisitely expressed by that exemplary expositor, the poet, Sumner A Ingmark: