Many casino buffs wonder where the gambling gurus get the house
advantages they blithely bandy about. And, since the authorities don’t always seem to
concur, how much of the information is hard fact as opposed to opinion or inference from
The figures espoused by ethical experts are obtained using the laws of
probability. You’ll occasionally see discrepancies. But these may not be true
disagreements. Rather, they may result from variations in precision of the calculations,
subtle differences among nominally similar games, and diverse definitions of edge.
Baccarat affords an example of the latter. Edge for this game is
specified in two alternate ways. Consequently, you may notice distinct sets of percentages
in two reliable sources. Yet, both may be correct and, ultimately, have equivalent bottom
I’ll toss you some numbers. They not only reveal how accurately
relevant quantities can be known, but also demonstrate the way games of chance are
described by math, not by guess ”˜n by gosh.
Baccarat involves two hands ”” Player and Banker. Solid citizens
bet on either one, or on Tie. Rules for dealing, standing, and drawing are predetermined,
so a computer can be set up to run through every possible way the cards in a shoe can form
a pair of hands, then tally how many go to Player, Banker, and Tie.
Such a program shows that eight decks form 4,998,398,275,503,366
(that’s four quadrillion, nine hundred ninety eight trillion, etc) pairs of baccarat
hands. Player wins 2,230,518,282,592,260, Banker wins 2,292,252,566,437,890, and
475,627,426,473,216 are Ties. Dividing each win figure into the total four-plus
quadrillion combinations gives the probabilities for Banker, Player, and Tie.
The percentages found by division could be stated with extreme
precision. But this would offer no practical benefit. I only gave lots of what the pros
call “significant figures” to stress the point about fact over opinion. For
casino gambling purposes, 44.62 percent for Player, 45.86 percent for Banker, and 9.52
percent for Tie have enough decimal places, maybe too many.
With this single set of win percentages, edge on Player and Banker can
be presented two ways. Both use the same arithmetic. Take the chance of winning times the
payoff, subtract the chance of losing times the bet, and divide the result by the wager.
Ordinarily, Player and Banker push when the hands Tie. Otherwise,
Player pays 1-to-1, and Banker pays even money minus a 5 percent vigorish for a net
operative payoff of 0.95-to-1.
Baccarat edge may be given as a fraction of the money effectively at
risk. This is what’s bet, not in all rounds, but in the 90.48 percent expected to
have a win or loss. It ignores money out of action in the 9.52 percent of rounds forecast
to push. The probabilities are adjusted accordingly, so 0.4462/0.9048 or 49.31 percent of
the decisions are predicted to go to Player and 0.4586/0.9048 or 50.69 percent to Banker.
On this basis, edge on Player is [(1)(0.4931) – (1)(0.5069)]/(1) = -1.38 percent, the
minus sign indicating that the house is favored. Analogously, edge on Banker is
[(0.95)(0.5069) – (1) (0.4931)]/(1) = -1.15 percent, the minus sign confirming
bias toward the house.
Edge on Player and Banker may also be figured by counting all rounds,
including pushes. With this definition, edge on Player is [(1)(0.4462) – (1)(0.4586)]/(1)
= -1.24 percent. Edge on Banker is [(0.95)(0.4586) – (1)(0.4462)]/(1) = -1.05 percent.
The two sets of figures match up when they hit the cash drawer, where
it matters most. The first accounts for 1.38 and 1.15 percent of about 90.5 cents per
dollar bet. The second represents 1.24 and 1.05 percent per actual dollar. After, say,
1000 rounds at $10 each, the house’s theoretical “take” either way is $124
and $105 on Player and Banker, respectively. The duality recalls the rhyming reverie of
the beloved bard, Sumner A Ingmark:
True understanding wipes out the confusion,
Of separate pathways reaching one conclusion.