# The numbers game

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Baccarat is the king of casino games in Macau. A staggering 88 percent of Macau casino action is baccarat, with all the other games combined accounting for just 12 percent.

Of that baccarat dominance, around 64% is VIP baccarat – played by the many high rollers who love trying their luck in Macau.

Why do we tell you this? Because we also know that baccarat players are fascinated by numbers and statistics! So we’ve decided to delve a little deeper into the maths behind this famous game. This isn’t an exercise in telling you how to beat baccarat, but it will provide some facts and figures to deepen your knowledge of the game. Enjoy!

Starting hands and naturals

We ignore the suits in baccarat, so there are 13 possible cards in the deck: 1 (the ace), 2, 3, 4, 5, 6, 7, 8, 9 and the four 0s (10, jack, queen and king). Whenever a card is drawn from the shoe, there is a 1/13th of it being a 1, 2, 3, 4, 5, 6, 7, 8 or 9, and a 4/13ths of it being a 0 (we are ignoring the minute changes that result from card removal for now). Because of the 13 denominations, the initial two-card hand of either the banker or player has 169 combinations. 25 of those 169 combinations result in a starting hand of zero (also known as baccarat). Each of the other totals possible (1, 2, 3, 4, 5, 6, 7, 8 and 9) has precisely 16 combinations, so they are all equally likely.

 Total Combinations Chance 0 25 14.80% 1 16 9.50% 2 16 9.50% 3 16 9.50% 4 16 9.50% 5 16 9.50% 6 16 9.50% 7 16 9.50% 8 16 9.50% 9 16 9.50% 169 100%

Notice also these chances for your first two cards:

 Chance of getting any natural (8 or 9) and winning 16.25% Chance of getting any natural (8 or 9) and pushing 1.79% Chance of getting a natural 8 and losing to natural 9 0.90% Chance of not getting a natural and losing to a natural 15.35% Chance of neither side getting a natural 65.72% 100.00%

You will notice that around one third of the hands are decided by a natural (that is, either banker or player or both get an 8 or 9), and about two-thirds of the hands continue on with neither side getting a natural 8 or 9.

Non-natural four card hands

What if neither side gets a natural? Well if both the player and the banker have 6 or 7, no more cards are drawn. Here are the chances:

 Player Banker Winner Chance 7 7 Tie 0.90% 7 6 Player 0.90% 6 7 Banker 0.90% 6 6 Tie 0.90% 3.59%

Given that the chance of either side having a natural is 34.28% and the chance of a non-natural four card hand (both sides have 6s or 7s) is 3.59%, we can conclude that the chance of any four card hand (a hand where the dealer only draws a total of four cards for player and banker combined) is 37.87%, and therefore the chance of a fifth card being drawn in the hand is 62.13%.

The fifth card
Let’s say there were no naturals and it wasn’t a 6s and 7s-only hand of just four cards. Now you know for certain there will be a fifth card (and maybe a sixth card) drawn. The maths starts to get really complicated here so the best thing to do is just show a summary of the chances of each player hand winning or losing:

 Player has Player wins Tie Banker wins Total 7 70.63% 17.57% 11.79% 100.00% 6 53.06% 17.57% 29.37% 100.00% 5 44.23% 12.78% 42.99% 100.00% 4 39.83% 12.09% 48.08% 100.00% 3 37.50% 10.02% 52.48% 100.00% 2 35.86% 9.33% 54.81% 100.00% 1 34.92% 8.64% 56.45% 100.00% 0 34.31% 8.30% 57.39% 100.00%

Of course these are the chances of the player winning from each starting player hand from 7 down to 0, given that a fifth card will be drawn and against a random banker hand. It would be too complicated to show each player hand against each banker hand but this definitely gives you a good start.

It’s interesting to note that you are never completely dead. Even with a player hand of 0 against the banker’s 7 (the worst possible situation outside of naturals), the player has a 15.4% chance of winning and a 7.7% chance of a tie, which is a 23.1% chance of getting out of jail. Not too bad!

Best bet
It’s pretty common knowledge amongst seasoned baccarat players that the banker is a slightly better bet than the player. But why?

Well, in an eight deck baccarat game there are an incredible 4,998,398,275,503,360 (that’s almost five quadrillion) different combinations possible of the 416 cards to give a 4, 5 or 6 card player versus banker hand. Of those nearly 5 quadrillion combinations, precisely 2,292,252,566,437,888 of them result in banker wins, 2,230,518,282,592,256 result in player wins, and a mere 475,627,426,473,216 result in ties.

Those enormous numbers are too hard to take in, but when you break them down to percentages, and factor in the 5 percent commission on banker wins, the bets come out like this:

 Banker Percentage Payoff Contribution Wins 45.86% 0.95 43.57% Loses 44.62% -1 -44.62% Ties 9.52% 0 0.00% 100.00% Edge -1.06%

 Player Percentage Payoff Contribution Wins 44.62% 1 44.62% Loses 45.86% -1 -45.86% Ties 9.52% 0 0.00% 100.00% Edge -1.235%

As you can see the banker bet is slightly better than the player bet, despite the five percent commission. Having said that both the banker and the player bets are amongst the absolute lowest house edge bets in the casino, and if you factor in the comps you earn through the casino’s player program and perhaps VIP rebates on the action, the house edge becomes minute. This is the major reason why baccarat is the game of choice for the high roller.

Many players don’t play by math and will freely ignore all of the statistics above, and of course that’s fine. However, it’s certainly interesting once in a while to take a look at some of the numbers behind this game.

### 1 Comment

• Tom Wenceslao says:

I’m not getting the numbers you posted in the paragraph "the fifth card". My calculations are way off from what’s shown in your chart.
The 70.63% Probability of "Player winning" if "Player has" a 7 as their first 2 cards is way too much… as is the 17.57% chance of a "tie" is also slightly high, and 11.79% for "Banker Winning" is too low.