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How Many Boxes Should You Cover with Place Bets at Craps?25 January 1999
Craps players often build strategies around "place" bets. These are wagers made directly on one or more of the numbers four, five, six, eight, nine, or 10. Experienced bettors recognize the symmetries in, and progression of, these options.
Dice devotees ardently argue over how many and which boxes to cover. One set of alternatives involves inside (six and eight), middle (five and nine), or outside (four and 10) numbers. Payoffs increase toward the outside, but odds against winning become steeper and the casino's inexorable edge gets more execrable. In sessions of reasonable duration, one pick is about as good as another. Personal preference, not to mention the fickle finger of fate, dictates whether you'll be happier with more numerous but smaller returns and lower house advantage -- or the converse.
Another choice facing place bettors is whether to concentrate a wager on one of a complementary pair of numbers, or split the amount between both. For instance, $10 to win $14 on the nine vs $5 each to win $7 on the five and nine. Some solid citizens prefer the larger returns associated with a single wager. Others would rather win half as much twice as frequently.
Does splitting or concentrating risk in this manner influence expected results? Losses when the seven appears are equal either way. Effective house advantages, likewise, are identical. But the bankroll fluctuations bettors experience decrease slightly as risk is split across paired boxes, so characteristics of games change somewhat following one or the other strategy.
To see the effect, imagine sessions in which place bets are made on 100 shooters. The bets are kept in action at the same level no matter how often they win, until the seven rolls and they lose.
Say that 1,000 players bet $12 on six (or eight) and another 1,000 put $6 each on six and eight. House edge doesn't vary, so the average loss across all players will be the same either way -- $34. However, computer analysis shows that differences in bankroll fluctuation or volatility affect the fractions of those who finish the sessions ahead or behind. Of the $12/six bettors, about 415 can expect to win and 585 lose; of the $6/six-eight bettors, about 395 can expect profits and 605 routs.
Here are figures for 1,000 players tossing $10 on five (or nine) and another 1,000 dropping $5 each on five and nine. After 100 shooters, the average expected loss is $66 either way. Of the $10/five bettors, 315 can expect to finish in the chips and 685 in the hole; of the $5/five-nine bettors, 300 can expect to savor the ecstasy of victory and 700 to suffer the agony of defeat.
Corresponding results are projected for 1,000 players wagering $10 on four (or 10) and another 1,000 placing $5 each on four and 10. After 100 shooters, the average expected loss is roughly $100 either way. Of the $10/four bettors, 250 can expect to be ahead and 750 behind; of the $5/four-10 bettors, 200 can anticipate glory and 800 ignominy.
The data show that shifting bets from outside to inside raises the likelihood of finishing the 100 hands with a profit. Within each level, single larger bets offer more chance of ending the session on a happy note. The higher volatility associated with concentrated risk also tends to reward winners with richer earnings. To be sure, there's a downside. Since the averages are the same, increased fluctuation produces more and bigger winners when the table is hot but induces heavier losses when it's cold.
As often in gambling, the decision is an individual matter. It affords you a way to tailor games to your fancies, going for the big score or moderating your risk. As Sumner A Ingmark, a bard who battles for balance in his ballads, bluntly blurted:
A gambler's lot becomes multidimensional.
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