Why It's Easy to Believe Rare Events Happen All the Time

Assume you go "with the dice" at craps. You start with a wager on the pass line, then likely add Odds and make Place, Buy, or Come bets after the come-out roll. Your worst nightmare is loading the layout, then watching the seven pop before you've collected anything. It happens. And when it does, you're entitled to mumble a few epithets. But you're supposed to grin and bear it, figuring you'll recover in due course. The story differs if you put money up and it all goes away with several shooters in quick order. You lose so much, so quickly, that recovery is difficult. Hopes of profit fade, and your goal switches to break even. Or maybe to a moral victory somewhere on the upside of your threshold of pain.

What are the chances of these awful scenarios unfolding? Forget, for a moment, what you may believe about bad shooters and cold tables. Instead, pretend you think the game is really random.

The likelihood that a shooter will establish a point and throw a seven on the subsequent roll is 24/36 multiplied by 6/36. This equals 11.11 percent. In round numbers, it's 11 chances out of 100 for any shooter. And the probability it'll take place several times in a row is 12 out of 1,000 for two, 14 out of 10,000 for three, and 15 out of 100,000 for four. Chances are lower of Come bettors getting up on three numbers in consecutive rolls, then striking out on the fourth. Probability is 2.675 percent. This is roughly 27 chances out of 1,000 for any shooter, 7 out of 10,000 for two in a row, and 2 out of 100,000 for a terrible trifecta.

These prospects are remote enough that up-and-down situations ought not arise especially often. Certainly not multiple instances in short order. So why does everyone remember how frequently they've endured these perturbing predicaments.

It's the perverse face of mind over matter. The triumph of impression over reality. Players tend to recall outstanding sessions, bad as well as good. Memorable events experienced on isolated occasions are just that. Memorable. And this imparts a perceived importance greater than their actual prominence.

There's more, too. Say that as the dice move around the table, a few shooters go up and right down. But imagine that nobody else is particularly productive -- maybe a sporadic repeat, perhaps a pass now and then, but nothing spectacular. It's easy to get disgusted and conclude that "everybody" is crashing on takeoff.

This view is supported by the statistics. Not the math of multiple events in succession. But the arithmetic of one or more appearances of ostensibly rare incidents within a relatively narrow interval.

For example, picture 14 solid citizens at a crowded craps table. The chance of any shooter setting up a point then bombing on the next roll is 11.11 percent. But, what's the probability of this transpiring one or more times as the dice move through the hands of the 14 rollers along the rail? It's 33.6 percent of one instance, 27.3 percent of two, 13.7 percent of three, and 4.7 percent of four. The chance that this fate will befall at least one of the 14 is almost 80 percent.

Similarly for Come bettors. The chance of three numbers being set up, bing-bing-bing, followed straightaway by a big bad bong, is only 2.675 percent. But there's a 26.3 percent probability it'll happen once among 14 shooters around the table, and a 4.7 percent chance of the lightning striking twice. The probability of at least one occurrence in the transit of 14 shooters is nearly 32 percent. Not overwhelming, but certainly a strong possibility.

Few craps players who bet "right" realize that spreading money over the felt is equivalent to making a "lay" bet. A good shot at winning less than the total at risk, offset by a small chance of a wipe-out, on any roll. So several hits are needed to compensate for the ultimate miss. Bear this in mind to stay ahead of the game. Emotionally, for sure. Financially? Well, here it helps to reflect on the rhyme of the rollers' Raleigh, Sumner A Ingmark:

Gamblers look at math suspiciously,
Casting blame about maliciously,
When the game goes unpropitiously.