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# Playing It Smart: Why you rarely get what you expect at craps

19 May 2008

Say you throw the dice randomly 36 times. According to the laws of probability, you'll expect to get six sevens, five sixes and five eights, four fives and four nines, three fours and three 10s, two threes and two 11s, and one two and one 12.

Payoffs for the various bets and characteristics of the game such as the house advantage, are based on the supposition that what you expect is what you get. Real life, even what passes in a casino for real life, doesn't work out quite this nicely. Which is why, in sessions or casino visits of reasonable duration, some solid citizens win big bucks while others lose their shirts.

One reason for the uncertainty is that the expectations aren't very strong. If you throw the dice 36 times, you may in fact obtain six sevens, five sixes, four fives, and so on. But you might also see none, or 36, or anything in between.

The chance of exactly six sevens is a mere 17.6 percent. The remaining 82.4 percent is the chance of throwing some other number of the li'l devils. An intuitive way to think about this is to picture 1,000 shooters throwing 36 times each. Were these percentages to hold for the 1,000 trials, only 176 would throw six sevens. The other 824 would get fewer or more. As examples, on the low side, 170 would shoot five sevens, 133 four, 80 three; on the high side, 151 would throw seven, 109 eight, and 68 nine.

So why do the math mavens tell you to expect six when they know in their hearts you can't reliably make book on it? For two reasons. First, six sevens is the average taken over lots of shooters. Second, it's the most likely result, as is evident from the way the figures drop above and below six sevens.

The effect is similar for all the numbers. The chance of either five sixes or eights is 18.9 percent. That of four fives or nines is 20.7 percent. Three fours or 10s have likelihoods of 23.4 percent. Two threes or 11s work out to be 27.8 percent. And one two or 12 ring in at 37.3 percent.

What happens when you go to more trials? Say 360 throws, when the expected number of sevens is 60? Intuition might suggest that the law of large numbers would increase the chance you'll hit the theoretical average. This turns out not to be the case.

The chance of 60 sevens in 360 throws is 5.634 percent, much less than the 17.6 percent of six sevens in 36 throws. But 60 is the average. And it's also the most likely result, with 5.634 percent greater than the probability of any other number of sevens. For instance, it's 5.615 percent for 59 and 5.541 percent for 61.

The way you may think the law of averages should work on larger samples seems to hold better comparing the likelihood of five, six, or seven sevens in 36 throws with that of 50 to 70 sevens in 360 throws. The former is 49.7 percent; the comparable range for 360 throws is a much greater 82.3 percent.

However, while more trials give you a better chance at being within a comparable range of the expected value, you're also apt to be numerically further away. Pretend every extra seven loses \$10. One seven above the average in 36 throws would set you back \$10; 10 sevens over in 360 rolls would represent a \$100 hit.

Another effect of more trials can be seen by comparing chances of fewer or more than the expected numbers of sevens in 36 and 360 rolls. With 36 rolls, probabilities are 43.1 percent of being under and 39.3 percent over. With 360 rolls, likelihoods are 47.8 and 46.6 percent under and over 60, respectively.

Food for thought, next time you're at a table and a bezonian at the other end drives the dealers nuts changing bets, muttering all the while that one out of every six rolls has to be a seven and the shooter has been throwing for a while so a seven's gotta be due. Better that you mind these mutterings of the immortal muse, Sumner A Ingmark:

The laws of probability should have been written stronger,

What I expect ain't happened yet and I can't wait much longer.
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Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.