Playing It Smart: How many throws does it take to resolve a bet at craps?

Nearly every craps aficionado has heard that "sevens show once every six throws," or words to that effect. If not actually scheduled like clockwork in any arbitrary set of six tosses, then at least expected in theory or found on the average. So, diligent dice devotees and other inquiring minds want to know why they get their ducks lined-up after the come-out, then are often knocked for a loop by a seven before anyone hoots or hollers even once.

Say the shooter has established a point and you've made a few Place bets. The chance of a wipe-out with a seven on the first point roll is one out of six 16.67 percent. The probability of anything else on the first toss with a seven on the second is 13.89 percent. Two and three other results, followed by sevens, come in at 11.57 and 9.64 percent, respectively. Likelihood of a seven on one of any four rolls exceed half, 51.77 percent.

Of course, optimism helps in gambling emotionally if not financially. Your rack can be half empty or half full.

Nervous Nellies may take bets down or off after five or six point rolls that yield no decisions, fearing the seven is due. More stalwart souls might be fortified in their long-haul faith by knowing the chance of a seven is still just 66.51 percent within six rolls, doesn't hit 90 percent until 13 tosses are considered, and reaches 99 percent only when agglomerating 26 throws.

A more positive question for solid citizens in whose breasts hope springs eternal involves the chance their bets will win in some number of throws before a seven spoils the fun. Make believe you have money solely on Pass. The accompanying table shows the cumulative chance of a seven before each point, and conversely of each point before a seven, for one to six throws.

Cumulative chance of rolling a seven or a specified point in one to six throws.

throws	point is 4 or 10	point is 5 or 9	point is 6 or 8
		seven	point	seven	point	seven	point
1	16.67%	8.33%	16.67%	11.11%	16.67%	13.89%
2	29.17%	14.58%	28.70%	19.14%	28.24%	23.53%
3	38.54%	19.27%	37.40%	24.93%	36.28%	30.23%
4	45.57%	22.79%	43.68%	29.12%	41.86%	34.88%
5	50.85%	25.42%	48.21%	32.14%	45.74%	38.11%
6	54.80%	27.40%	51.48%	34.32%	48.43%	40.36%

s you might anticipate, the cumulative chance of either the point or the seven increases with more throws. Proportions, though, stay constant. To illustrate, the ratio for the chance of a seven before a five within three throws is found from the data as 37.40/24.93, which equals 1.5. Within six throws, the ratio is 51.48/34.32 which likewise equals 1.5. Perceptive punters will recognize that the value is the same as 6/4 combinations that yield a seven divided by ways you can form a five. You can envision the ratio as odds against winning 6-to-4 or 1.5-to-1.

The sum of the cumulative chances over any series of throws is the probability that the bet is settled either way. To illustrate, combined prospects of winning or losing on a six within five throws are 45.74 + 38.11 or 83.85 percent. You exceed the 50 percent chance of a decision on the six at two throws.

Craps players typically have multiple numbers in action on point rolls. More numbers give more winning combinations. For example, say the point is four and you Place the eight and nine twelve ways to succeed. The chances on the first roll are 33.33 percent (12 out of 36) of one of the numbers and 16.67 percent (six out of 36) of a seven. The ratio is 2/1, odds of 2-to-1, that one or another of the bets will win before everything goes kablooie. The outlook for victory or defeat within five rolls turns out to be 64.58 or 32.29 percent, respectively. Much stronger for joy than sorrow, although the odds stay at 2-to-1. And a loss sets you back more than a win gains the offset averaging the 2-to-1 ratio plus a factor to account for house advantage.

Oh. About those sessions when shooter after shooter establishes a point then crashes immediately. The chance of two such zonkers in a row is 2.78 percent, three is 0.46 percent, four is 0.08 percent, and so on. The more the lower. But don't be shocked when it occurs. Instead, remember this, by the poet, Sumner A Ingmark:
Although a chance may be quite small,
It's bigger than no chance at all.