Calculating House Edge In Craps

By John Grochowski on Thursday May 19, 2016

bets, betting, betting-strategy, casino, casino-games, casinos, chance, chances, craps

I don’t like to get bogged down in math in this blog. All casino games have roots in math, but you don’t have to be able to do the math yourself to shop for the best bets. House edges are easy to find online or in books about gaming.

Nonetheless, I do get reader requests for explanations about the underlying arithmetic, especially in craps.

“Can you explain how house edges are derived?” one reader asked. “Maybe you could walk through the math for the house edge on the pass line.”

The house edge on the pass line is one of the more involved calculations in craps, because pass is a multipart bet. You have to account for both comeout roll wins on 7 and 11 and losses on 2, 3 and 12. You also have to account for the wins and losses after a point is established.

For the first part of the bet, you need to add the probability of rolling 7 to the probability of rolling an 11. Since there are 36 possible two-dice combinations, with six totaling 7 and two totaling 11, we’re adding 6/36 to 2/36, giving us 8/36. We’ll need that total later.

For the second part of the bet, the formula is pr(4)×pr(4 before 7) + pr(5)×pr(5 before 7) + pr(6)×pr(6 before 7) + pr(8)×pr(8 before 7) + pr(9)×pr(9 before 7) + pr(10)×pr(10 before 7). The “pr” stands for “probability,” so the long form is “the probability of rolling a 4 times the probability of rolling a 4 before rolling a 7,” and so on for each point number.

I’m going to take a shortcut here and just tell you the answer is 9,648/35,640. We need to add that to 8/36 to get our total chances of winning, and that comes to 17,568/35,640.

What that tells is that for every 35,640 trials, we win 17,568 times and the house wins 18,072. The house wins 504 more times than we do. Divide those 504 by the 35,640 trials, then multiply by 100 to convert to percent, and you get 1.41 percent.

That’s the familiar number you’ll find on any list of craps house edges. And really, it’s fine just to use the lists. I promise not to be so calculation heavy again – unless someone really wants to know.