The Chance of a Truly Average Game Is Less than You May Think.

9 November 2005

By Alan Krigman

Huey, Louie, and Prudence are apocryphal nickel craps buffs. They all put $5 on Pass, then make one additional minimum Place bet on the high side if the point is low, or vice-versa.

Here's where they differ. Huey likes the six or eight because the edge is lowest; his $6 bets pay $7 and give the house only 1.5 percent juice. Prudence prefers the four or 10 since they return the most; her $5 wagers earn $9 and she doesn't care what the gurus say about a 6.7 percent penalty she can't see. Louie, a fan of Goldilocks, picks the five or nine; he figures that winning $7 on a $5 wager with a 4 percent house advantage is "just right."

The stalwart three argue endlessly about why they bet what they do. To settle the issue, they thought about keeping records of their sessions when they played at the same table, to see who came out best. Would the results be conclusive?

To picture what some think the math decrees will happen, consider a session of about two hours' duration in which Place bets are in action for 180 throws. The laws of probability say that a seven is expected 30 times. What if probability were certainty? On six or eight, Huey would win 25 times, picking up $7 x 25 or $175 and dropping $6 x 30 or $180 for a net loss of $5. On five or nine, Louie would score 20 wins, having revenues of $7 x 20 or $140 and costs of $5 x 30 or $150 for a net loss of $10. And, on four or 10, Prudence would taste 15 successes, finishing with $9 x 15 or $130 income and $5 x 30 or $150 expense for a net deficit of $15. On this basis, a few sessions' data would show edge as governing.

Of course, if averages applied definitively to every situation, the casinos would be empty. Nobody would make bets confident of losing an amount equivalent to the edge on their action. Instead, bettors hope for sessions in which they win more or lose less often than an ingenuous interpretation of statistics suggests.

To get perspective on how this works in practice, think about 36 throws. The probabilities imply six sevens, five sixes and eights, four fives and nines, and three fours and 10s. With these frequencies, nets for the sequence would be losses of $1 on six/eight, $2 on five/nine, and $3 on four/10. However, the chances of rolling exactly the averages turn out to be small.

What happens in 36 throws if instances of winning numbers are as expected but just five sevens appear? Now, the sequence is profitable netting $5 for six/eight, $3 for five/nine, and $2 for four/10. Similarly, if six sevens pop during the 36 throws but each Place number shows one time more than the average would have it. Profits are then $6 for six/eight, $5 for five/nine, and $6 for four/10. Probabilities for these cases are compared with the statistically correct values in the accompanying table.

Probabilities of the combinations of sevens and points close
to the statistically correct number in 36 throws

point	one less seven	statistically correct	one more number
4/10	4.00%	4.15%	2.57%
5/9	3.55%	3.69%	2.95%
6/8	3.25%	3.38%	2.81%

The data confirm that the statistically correct combinations of wins and losses, leaving the player in the hole, are the most apt to occur for each Place bet. However, the probabilities are only slightly more than those of one less seven or one excess number, each of which would make the sequence profitable. Chances of situations in which the loss would be worse than the theoretical average, for instance an extra seven or one less number, are similarly close to those of ending square on the average.

The chances of various specific combinations of wins and losses even during moderately short sequences are low, and small departures aren't much less likely than the theoretical expected values. This explains why gamblers find from experience that house advantage doesn't prevent disciplined, or lucky, solid citizens from having lots of winning sessions. It also helps understand why a few months of records have too much variance for Huey, Louie, and Prudence to definitively resolve their alternate perspectives. As the punter's poet, Sumner A Ingmark, put it:

Misconstruing statistical facts,
Fosters belief in mystical acts.