In the episode we recorded last night (#219, still unpublished as of the time of this post) I made a bit of a mess of the explanation on how to come up with the pass line odds on crapless craps, and incorrectly declared that it was easier to do than normal craps. Anyway, it’s a fun little math exercise, so let’s quickly do it.
| Come out Roll: | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Sum | (1 – Sum) – Sum |
| Odds of Happening | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 | 1 | |
| Odds of Winning Given it Happening (Regular) | 0 | 0 | 3/9 | 4/10 | 5/11 | 1 | 5/11 | 4/10 | 3/9 | 1 | 0 | ||
| Odds of Happening * Odds of Winning | 0 | 0 | .0278 | .0444 | .0631 | .1667 | .0631 | .0444 | .0278 | .0556 | 0 | .4929 | .0142 |
| Odds of Winning Given it Happening (Crapless) | 1/7 | 2/8 | 3/9 | 4/10 | 5/11 | 1 | 5/11 | 4/10 | 3/9 | 2/8 | 1/7 | ||
| Odds of Happening * Odds of Winning | .0040 | .0139 | .0278 | .0444 | .0631 | .1667 | .0631 | .0444 | .0278 | .0139 | .0040 | .4731 | .0538 |
I’ve quickly made a chart above that shows the odds of winning a passline bet for both regular craps and crapless craps. These numbers are slightly rounded for simplicity, but if you add the unrounded numbers you should get the exact odds of the player winning any given pass line bet, and thus house edge. Let’s quickly walk through these.
The first number that matters are how likely it is to roll any of these given numbers. There are 36 combinations of ways two dice can come up, and only one of those (1,1) is 2, and six of them (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) are seven. Just counting these up is how to get the odds of these rolls happening on an initial come out roll.
We take that number and multiply it by the chance you win the bet based on that initial roll. In both craps and crapless craps, that number is 1 (100%) if you roll a seven, since that is just a winner. In regular craps 11 is also a guaranteed winner, while 2, 3, and 12 are guaranteed losers. For the other numbers, the only rolls that matter are the odds of hitting the initial number again vs the odds of rolling a seven, since all other rolls don’t impact your pass line bet. If you set a 4 as the point, there are 3 possible rolls of the dice that make your pass line bet a winner (1,3), (2,2), (3,1), while there are always six ways to make a seven. So there are nine total rolls that resolve the bet, and you win three of them.
To come up with the total likelihood that the player wins their pass line bet you just add up the odds that the player wins each individual number times the likelihood that number comes up. For both regular and crapless craps the pass line will win just over 43.7% of the time from just initially rolling the numbers between 4-10 initially, the only difference between the games is on the numbers 2, 3, 11, 12.
In normal craps 2, 3, and 12 are automatic losers, and 11 is an automatic winner. This adds a full 2/36 odds (just over 5.55%, the odds of rolling an 11) to winning the bet, pushing the likelihood of a player win to 49.29%. In crapless craps you get a chance to win initial rolls of 2, 3, and 12, but you have to work to win an 11. You end up gaining around .0218 extra wins from rolling a 2, 3, and 12 on the come out roll in crapless craps, but you end up losing .0416 wins from not automatically winning on an 11, and having only a 1/4 chance of winning an initial point 11 compared to 100% chance in normal craps. This makes the player win only 47.31% of pass line bets in crapless craps.
So it’s the 11 that is why the house edge of a pass line bet in crapless craps is around 5.38% while it’s only 1.42% for normal craps.
It is really fun to get a huge odds payout on a craps number in crapless craps though.