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Casino Math

Craps

The key to understanding the math of craps is understanding the dice combinations, or odds. In my strategies, we only want to play the bets that have the best probability of winning. These are the pass line at odds, bets at odds, occasional bets on 6 and 8, don’t pass setting odds, and don’t come with or without weighting odds.

If you play the bets mentioned above, the house percentage in craps is the lowest of all casino games. Taking single odds on the pass line and future bets reduces the house percentage to 0.8%…double odds reduces it to 0.6%…triple odds reduces it further to 0, 5%…and 10 times up to 100 times the odds makes the game almost dead even.

At seminars, I am always asked why the bets placed are not as good as the bets come. The answer lies in dice combinations. A placed bet can be used to illustrate this point. A bet placed directly on the number 5, for example (i.e. a place bet), can only win on a total of four dice combinations: 1-4, 4-1, 2-3, 3- 2. That’s it! When a 7 is rolled, which represents a total of 6 dice combinations, the bet loses. It’s 6 vs 4, or 3 vs 2 against you based on dice combinations alone.

Now let’s look at a come bet. When the come bet is in the come zone, it wins on a seven or 11 for a total of eight dice combinations and loses on a 2, 3 or 12 for a total of 4 dice combinations. It’s 6 to 4, or 2 to 1 in your favor for immediate victory against immediate loss. If this come bet should go to 5, for example, he now has 4 other combinations of dice to win. So the come bet that started in the come zone and went to 5 had 12 dice combinations up for grabs, compared to only 5 combinations for the placed bet on 5. That’s a huge advantage. This analysis can be applied to each bet placed.

Adding the fact that you can take odds on all come bets, the casino advantage on bets placed on 4 or 10 is 6.7%; on bets placed on 5 or 9, it is 4%; and placing bets on the 6 and 8 is 1.5%. A come bet, whatever number it goes to, is only 0.8% at single odds, exactly the same odds as the pass line at single odds.

To win at craps, you need to minimize the house edge and use money management to capitalize on all streaks, do or don’t. That’s what Benson strategies are for.

black jack

Blackjack is the only casino game where the player’s advantage or disadvantage changes with each card played. The game itself favors the house by 4%, mainly because if you bust and the dealer pauses, guess who gets the money? The house, of course!

This house edge can be reduced to 1.5% by playing basic strategy. That in itself makes it a good game to play. With proper base play and good money management, you can expect to show a positive return over time.

Additionally, tracking played cards, combined with basic strategy, can change a player’s edge by 1%. The player’s advantage increases as more high cards remain in the unplayed deck (or shoe). High cards favor the player because they give him a better chance of getting a “pat” hand and they also increase the chances of the dealer busting. The dealer must hit 16 or less. With high cards remaining, this creates a greater chance of a dealer break.

The most common methods of tracking are simple hi-lo counts (good on single deck games) and card grouping methods (good on shoe games). A 1% edge means expertly played blackjack is the only casino game that offers the player an expected positive mathematical return.

Baccarat

Baccarat is known as a game of negative expectations (the same as craps, roulette and others). This means that the odds always favor the house. By always, I mean that there is no known method of play that will mathematically place the odds in favor of the player. This can only be done with perfect card counting in blackjack (that’s why, of course, they don’t allow you to win much).

The way we win at baccarat is to follow the trend. A trend will develop in any series of random or near-random events. Remember that you won’t have enough time to establish actual probability numbers, as these depend on a lot of play to reach statistical significance. You could be biased one way: 50% more players than bankers for example (which would be great, by the way).

The casino sees real statistical significance because there is so much action going on all the time. They cannot lose from the game itself. They can only lose if they don’t have enough players or in typical trading profit/loss scenarios. But they don’t LOSE on the game itself. It is not possible. But it is very possible that the casino will lose to individual players. The casino compensates for these losses because it has enough players to make the math work for them in the long run.

This last point is very important. Because unless you play around the clock, you will never be playing by the same mathematical statistics as the casino. Our starting rules and money management eliminate this immediately. The casino will only beat the Baccarat player for lack of discipline and/or poor play.

Roulette

Roulette has a 5.26% advantage over the player. The reason for this is that there are actually 38 numbers on the wheel: 1-36 and 0 and 00. The winnings, however, are based on the 36 numbers only, not the 0 and 00. The single number pays 35-1. So, in simple terms, the 0 and 00 are the house edge.

Over a long period of time, the casino will have a definite mathematical advantage.

Casino math

  • Needs a lot of game action to get real odds.
  • All statistics are based on an infinite number of throws.
  • Hate discrepancies in bet sizes.
  • Dislikes structured play, especially in starting rules and money management.
  • Once the game volume is reached, the mathematical advantage is guaranteed.
  • The casino will offer any incentive to achieve this guaranteed mathematical advantage.

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