Let’s say that you are playing craps and make a bet on the pass line. The shooter establishes a point during the come-out roll and you take the odds with no other wagers. How many throws can you expect to come before a decision, either the point is repeated or the ugly seven shows up?
A seven can be made with six of the possible 36 combinations the dice can form. A four or 10 can be formed three ways each. If either of these numbers is the point, the chance of settling the bet on any roll is then nine out of 36 or one out of four. So the theoretical average is four rolls per decision. Similar reasoning yields 3.6 and 3.27 rolls per resolution on five/nine and six/eight, respectively.
You undoubtedly remember lots of occasions when the decision came right back at you one way or the other. And, you’ve probably experienced games in which the shooter seemed to roll endlessly before finally making the point or sevening-out.
Computer simulation yields some ballpark figures that can help provide you with insights. For this purpose, 10,000 virtual solid citizens were programmed to hurl those hexahedrons for points of four or 10, five or nine, and six or eight. The numbers of shooters resolving each point by hitting or missing are shown in the accompanying table. The values would differ slightly, up or down, in other runs, that’s the nature of simulation as opposed to analysis.
Throws to resolve a point at craps
based on 10,000 simulated shooters
Throws Instances
4 or 10 5 or 9 6 or 8
1 2,428 2,737 3,095
2 1,862 2,123 2,097
3 1,520 1,454 1,516
4 988 1,031 961
5 778 746 771
6 659 560 474
7 477 369 303
8 302 263 227
9 266 220 180
10 169 125 99
11-20 524 367 267
over 20 27 5 10
The averages for the virtual shooters are close to what’s theoretically “expected” from the laws of probability. A surprise is that the expected values aren’t those that occur with the greatest frequency. Regardless of the point, a decision on the first roll turns out to be most common. Further, for each point, well over half the bets are resolved within the first three throws — that is, in fewer than the expected number. These quickies are offset over the long term by the few shooters who go for extended periods before hitting the point or a seven.
Of course, true craps aficionados generally have multiple numbers working during a hand. Three is typical, for instance a point of six and Place bets on five and eight. In this case, decisions will be reached in any of six ways for the seven and 14 ways for the indicated numbers, 20 out of 36 in all. The average would be once every 1.8 throws. Of the 10,000 virtual shooters in this situation 5,487 hit one of the numbers or a seven on the first roll, followed by 2,501 on the second, and 1,091 on the third. The maximum number of rolls before a decision in this particular simulation run was 11, but in only five out of 10,000 instances.
Exclamations often heard in the craps pits are (expletives deleted) “unbelievable!” and “this is the worst table I’ve ever seen!” It may seem so, after awaiting a decision for what you think is a record number of throws, when you finally eke out a win or crash ignominiously. However, the simulations show that while it’s easy to become accustomed to the quick turn-arounds that happen regularly, long delays aren’t really remarkable.