The house edge on the don't pass

Craps can be a game of opposites. You can bet the shooter's going to make his points on pass and come, or bet he won't make 'em on don't pass or don't come. You can place the 6, betting the shooter will roll a 6 before a 7, or lay the 6, betting the 7 will come first.

And so on.

A reader who identified himself as Jason phoned last month, asking about don't pass.

"It's the opposite of pass, right?" he asked. "So if the house has an edge on pass, why don't I have an edge on don't pass?"

One tiny change is enough to do it. If the shooter's comeout roll is a 12, pass bets lose, but don't pass bets don't win. They just push, giving don't pass bettors their money back. That tips the balance so that the house has a 1.36% edge on don't pass, compared with 1.41% on pass.

Jason asked if I could go into more detail, which was really too detailed for a phone conversation. But let's imagine a sequence of 1,980 come out rolls in which each possible combination of two dice comes up 55 times and all outcomes stick to true odds. And let's say a player is betting $10 on don't pass on each comeout, risking a total of $19,800.

• 165 comeout rolls are either 2 or 3. The don't pass bettor wins $10 on each, or $1,650, and keeps the $10 wagers, for another $1,650.
• 440 comeout rolls are either 7 or 11. Don't pass bettor loses $10 on each.
• 55 comeout rolls are 12. Don't pass bettor pushes, and just gets the $10 wagers back, for $550.

Through those 660 rolls, the don't pass bettor has put $3,850 back in the rack.

The other 1,320 comeouts are point numbers.

• 165 comeout rolls are 4, and 165 are 10. Don't pass wins 110 times on each -- 220 wins resulting in $2,200 in winnings, and the bettor keeps the $2,200 on those wagers won.
• 220 comeout rolls are 5, and 220 are 10. Don't pass wins 132 times on 5 and 132 on 10 — 264 wins for $2,640 in winnings and $2,640 in wagers kept.
• 275 comeout rolls are 6, and 275 are 8. Don't pass wins 150 times on 6 and 150 on 8 — 300 wins for $3,000 in winnings and $3,000 in wagers kept.

Add all that up, and after decisions following all 1,980 comeout rolls, the don't pass bettor has $19,530 left from the $19,800 in wagers. The house has $270. Divide the $270 house profit by $19,800 in wagers, then multiply by 100 to convert to percent, and you get the house edge: 1.36%.

In real life, there's no such thing as a perfect sequence that plays out strictly by the odds. Sometimes you'll win, sometimes you'll lose, but in the long run the house will collect something very close to 1.36%.

ON THE EDGE: In some gambling books, you'll see the house edge on don't pass listed as 1.4% rather than 1.36. It's not a matter of rounding. Those that list the higher edge assume that if the shooter rolls a 12 on the comeout and the bet pushes, you won't just pick up your bet. They assume you'll replay that money until there is a decision, and if another 12 is rolled on the next comeout, you'll replay again, and on and on until you either win or lose. The effect of replaying those pushes boosts the house edge to 1.4%.

I used to be in the camp that listed the 1.4% edge. Over the years, I came to the conclusion that a push is a valid decision. There are pushes in blackjack when your 17 pushes a dealer's 17 — or 18 vs. 18 on up to 21 vs. 21.

There are even pushes in video poker, though we don't call them that. When you get a 1-for-1 payoff, such as a pair of Jacks in Jacks or Better-based games, it's labeled a winning hand, but you're just getting your money back. That's a push.

After a push, you have the option of picking up your money and leaving. The casino considers it a legitimate decision on a bet, and so do I.

In the explanation for how the house edge is derived on don't pass, treating a push as a legitimate outcome means we could reverse the outcomes on the same number of rolls to explain how the house edge is derived on pass. If we assumed extra bets after pushes, I'd have to use more rolls to explain don't pass than to explain pass.

So for me and don't pass, 1.36% it is.