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A Quick Math Lesson on Craps, Casino Edges

7 July 2006

A reader has asked me to explain how "they" come up with a casino edge of 9.09 percent on a craps hardway 6 or 8 and 11.11 percent on a hardway 4 or 10. (I should probably warn you at this point that obviously there's some math in the explanation, but trust me, it's not hard to follow.)

But first a short explanation of hardway bets. The hardway bets are in the middle of the table and pay 9-to-1 for a hard 6 or 8 and 7-to-1 for a hard 4 or 10. When you make a hardway bet, you are betting that the shooter will roll an even number, a 4, 6, 8 or 10, as a double either before a 7 is rolled or before the number is rolled the "easy" way.

For example, if you bet on the hard 8, there is only one way for the hard 8 to be rolled: 4-4. But there are four ways for an easy 8 to appear: 6-2, 2-6, 5-3, 3-5. Plus, there are six ways for a 7 to be rolled. So that means a total of ten ways to lose and one way to win. Correct odds are 10-to-1, but the payoff is only 9-to-1.

Now for the definition of house edge. I like the way Fred Renzey, author of several gambling books, states it: "House edge is the casino's profit divided by the player's total bets." That's it; it doesn't get any simpler than that.

So, if you make 11 $5 bets in a row on a hard 8, you will win one time out of 11 and will lose 10 times out of 11. You will win $45 the one time you win because of the 9-to-1 payoff, and you will lose a total of $50 for the other 10 times. The casino's profit is $5, and you have bet a total of $55. $5 divided by $55 is 9.09 percent.

Go through the same process with a hard 6, sister number of the 8. There are four ways to make an easy 6, six ways to make a 7, thus ten ways to lose. Only one way to win it, by rolling a 3-3. Correct odds are 10-to-1, but payoff is only 9-to-1. Again, house edge is 9.09 percent.

The hard 4 and hard 10 are even worse. There are only two ways to roll an easy 4 or easy 10 and still six ways to roll a 7. So the correct odds are 8-to-1 against a hard 4 or 10, but the payoff is only 7-to-1. Out of nine rolls, you've won $35 and lost $40. The casino's profit is $5 out of the $45 you've bet. Divide $5 by $45 and you come up with 11.11 percent.

We call this percentage the house percentage or house edge. Apply that edge to each bet made and you have the average dollar amount the casino expects to win, or, for example, $9 from every $100 bet on the hard 6.

Needless to say, dice don't read the same book that experts write. They don't know that they're supposed to form a hard 6 one out of ten times. That's how you get "hot" tables, where a series of good rolls comes up more often than a streak of bad ones.

Unfortunately, it's also how you get "cold" tables, where the easy numbers and 7s outnumber the hard numbers. Trust me, it balances out and, in the long run, the casino will get more than it's share.

Casinos know this, and that's why casino executives love to see hot tables and craps players hootin' and hollerin' and lovin' this game. Because the execs know that the players will return to give back the money they've won.

Until next week, dice be nice.

Linda Mabry

Low Roller Linda Mabry lives and gambles on the Mississippi Gulf Coast. She writes a weekly, general gambling advice column for the Biloxi Sun Herald, and may be contacted through her e-mail address, lnmabry@cableone.net or her web site www.thelowroller.com
Linda Mabry
Low Roller Linda Mabry lives and gambles on the Mississippi Gulf Coast. She writes a weekly, general gambling advice column for the Biloxi Sun Herald, and may be contacted through her e-mail address, lnmabry@cableone.net or her web site www.thelowroller.com