A pass is an initial roll in craps that either wins right away or establishes and makes a point. A come-out roll (first) of a 7 or 11 results in an immediate win; a come-out roll of 2, 3 or 12 results in an immediate loss. If nothing of these options appears, a point is awarded. A point is the sum of the first four, five, six, eight, nine, or ten numbers rolled. The player (or shooter) continues to roll the dice until the established point is reached (the player makes the point) or a 7 is rolled. If a player craps out (or sevens out), the dice are passed to the next player to begin a fresh run. A pass is a successful outcome (such as throwing a 7 or 11 right away or establishing and then making a point).

## The Gambler’s Fallacy is now complete.

There are two types of gambler’s fallacy. A succession of events will continue, according to one. On the craps table, if a player makes multiple passes in a row, it is expected that the player will continue to make passes. Or, if red appears on a roulette table numerous times in a row, red should appear on the next spin, thus it is the moment to bet on red. The other version of the gambler’s fallacy holds that a succession of occurrences makes it more probable that the streak will come to an end. If red appears on a roulette table numerous times in a row, black is more likely to appear on the next spin. The underlying “logic” behind the second argument is that events with a 50-50 chance of happening tend to “even out” over time, so if red appears 10 times in a row, black is “due.” It is undoubtedly true that the number of heads and tails will be equal over a very long sequence of, say, tosses of an unbiased coin (one that is equally likely to show up heads or tails). However, assuming that this is the case in a small series is a mistake. The probability of heads on a single throw is 50%. If heads comes up ten times in a row, the next toss still has a 50 percent probability of coming up heads. Coins don’t remember anything. Dice and roulette wheels aren’t any different. Each event takes place independently of the others and is unaffected by previous events.

## Math

The house edge on a pass line bet is roughly 1.414 percent, so the likelihood of making a pass is about 0.493 percent. Making 15 passes happens roughly 0.0000247 of the time, with odds of 40,485 to 1 against. Making 25 passes happens roughly 0.0000000209 of the time, with odds of 4,7733,911 to 1 against. It’s unusual, but it does happen. The longest streak ever recorded was 154 passes. The chances of such happening are about 2.003 x 1047 to 1, which is a little higher than the number 2 followed by 47 zeros. That rare incident is unlikely to have occurred more than once in all of the world’s casinos since they opened, both in physical locations (brick-and-mortar casinos) and online.