Taking Odds in Craps Goes Beyond Cutting House Edge

Most dice devotees bet the odds because they've been told it's the correct thing to do... the only way to make real money at the game. But few players really understand why. Why it's preferable to just betting more from the get-go. Why a casino offering five or 10 times odds is giving solid citizens a bigger break than the joint down the alley with measly double odds.

So bones buffs at $5 tables offering triple odds often bet $10 on the line and take $10 odds rather than $5 and $15. Or, at $10 tables offering double odds, toss $10 on pass and take $20 odds, then bet numbers for $30 rather than $10 come with $20 odds.

Gambling knows no absolute right or wrong. Alternate strategies affect the probabilities of winning or losing various amounts, long or short term, and change the house advantage on the action. But each result is a matter of chance, and when a sucker bet hits for big bucks, who's gonna call the winner stupid for trying?

For instance, $30 bets on the pass line can win more than $5 flat with $25 odds. All it takes is a few shooters throwing strings of sevens and 11s on come-out rolls, then missing their points.

The usual rationale for raising the odds relative to the flat part of a line or come bet is that it lowers overall house edge on the wager. This is true. A player with $15 on the pass line has an expected loss due to the house edge of $21.21 for every hundred bets. A player spreading the $15 as $5 on the line and $10 odds cuts the expected loss to $7.07 per hundred bets.

Most gaming gurus fudge the fact that edge is rather small, either way. Distinctions between theoretical losses of $0.21 on a $15 line bet and $0.07 on a $5 line bet with odds are only meaningful to casinos, which log billions of wagers. They are essentially irrelevant to individual players over even multiple marathon sessions. One $15 payoff more than statistically predicted in 100 come-out rolls is worth more than the $14.14 extra theoretical loss during the hour of associated play.

Still, maximizing the odds-to-flat split has benefits. They're due to the effect on the amounts and probabilities of various wins and losses, as illustrated in the accompanying diagrams.

The first figure shows the theoretical distribution of results for a $15 flat bet. Here, a player can expect to win and lose $15 with roughly equal probability -- 49.3 percent chance of winning versus 50.7 percent chance of losing. The distribution is symmetrical and exhibits approximately zero skew.

The second figure depicts amounts and probabilities of wins and losses for a $5 line bet and $10 odds. With this bet, low-probability high-end $25, $20, and $17 wins counterbalance a somewhat greater chance of more moderate $15 losses. Expected results are stretched out on the positive side and clumped on the negative. The distribution has a skewness of +0.3. With five or ten times odds, the distribution would stretch even more toward the positive side and characteristic skewness would be greater.

In games with zero skew, players' bankrolls are likely to rise and fall gradually. As skew grows increasingly positive, bankrolls tend to decline steadily over longer intervals and escalate rapidly during short intermittent bursts. Players with the resources, time, and discipline to outride the downswings typically favor strategies exhibiting moderate positive skew because these games afford reasonable opportunities to surge ahead in spite of frequent and sometimes prolonged cold streaks.

The punters' poet, Sumner A Ingmark, skoffed at those who scorn skewness in games of chance when he wrote: