The math of the pass line

24 September 2017

By John Grochowski

Connie was a student at a daylong seminar I gave for a park district group around 10 or 12 years ago. I taught the basics of how to play casino games, and Connie discovered she loved craps.

“I could never figure it out before that day,” she told me later. “It confused me. How could 7 be a winner sometimes and a loser others? But now it’s my game.”

I’ve received a number of e-mails from Connie since then, always about craps. The latest had to do with the house edge on the pass and odds combination.

“I just take it as read that the house edge gets lower as more of your bet is on the odds. I still have a chart at that seminar where you show us the edge on the pass bet is 1.41% with no odds, becomes 0.8% with single odds, 0.6% with double odds and so on.

“It gets lower and lower and lower, but you didn’t tell us when it becomes even, and nothing I can find shows a break-even point. So tell me, is there a break-even point at million-times odds or something?”

Alas, the answer is no, there is never a break-even point.

Pass plus odds is a combination bet, just like placing 6 and betting on any 7 would be a combination, or any other grouping of multiple bets would be a combination.

The house edge of a combination bet always is a weighted average of the component bets.

In the pass-odds combination, one component is pass, which always has a 1.41% house edge. The other component is the odds, which always has a house edge of zero.

Zero plus 1.41 equals 1.41. If you were making two equal bets and they both were in action for the entire time, you could just take that 1.41, divide it by two and get a house edge of 0.7%.

But you can’t make the odds bet until the shooter has established a point. And when the shooter has established a point, the house edge is higher than 1.41%. Players have an edge when on the comeout, when the six ways to make 7 and two ways to make 11 are winners while only the one way to make 2, two ways to make 3 and one way to make 12 are losers.

The pass bet itself can be seen as analogous to a combination bet, with a player edge on one roll, a house edge on the rest, and an overall house edge that is a weighted average of all rolls.

Since you can take the odds only during the portion of the bet when the house has a higher edge, the weighted average of pass plus single odds is 0.8%, not 0.7. With double odds, you would count the house edge on the odds bet twice and the pass bet once before averaging. With 100x odds, you would count the edge on the odds 100 times and that on pass only once before averaging – essentially, the result is the edge on pass after the comeout divided by 101.

So we see ever declining house edges as a bigger percentage of the bet is on the odds. At double odds, the house edge is 0.6%, declining to 0.4% at 3x, 4x, 5x odds, 0.3% at 5x odds, 0.2 percent at 10x odds and 0.02% at 100x odds.

No matter how many times odds are offered, the 1.41% edge on pass remains part of the equation in calculating the weighted average. So the overall edge on the pass-odds combination can never reach zero.


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