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# How Spreading Out Bets Affects Gamblers' Prospects

20 September 2004

Lyle and Lydia have a running argument about craps. Lydia likes to bet \$82 "no four;" she nets \$38 if the dice come up seven and loses the \$82 if four pops. Lyle prefers \$41 each "no four" and "no 10;" he also nets \$38 on a seven, since he gets \$19 for each of the two bets, but only loses \$41 if fate furnishes four or 10.

Is either approach superior? Say two people played for equal lengths of time, with one or the other option the only bets they made. The house would rate their action as identical. But their gambling experiences would differ.

The casino essentially considers sets comprising 36 statistically correct rolls each. In such a set, Lydia would win \$228 (\$38 x 6); she'd lose \$246 (\$82 x 3); she'd finish \$246 - \$228 or \$18 in the hole. Lyle would win \$228 (\$38 x 6); he'd lose \$246 (\$41 x 6); he'd also end down by \$246 - \$228 or \$18. Both give the house a theoretical \$0.50 per roll.

This analysis ignores volatility and skewness. These shape the results of play in the short span of one session or casino visit and -- it turns out -- over lengthy periods as well. In terms of volatility, both bets have a 6/36 likelihood of a \$38 win but the single-number wagers are more volatile because the 3/36 chance of \$82 losses implies fewer but larger bankroll fluctuations than the 6/36 shot at \$41 hits. Similarly, the single-number wagers are more highly skewed, giving smaller chances at bigger swings.

A computer simulation illustrates how gamblers' fortunes evolve in practice. The alternatives were evaluated for 5,000 players after 300, 3,000, and 30,000 rolls, with the bets in question being the only wagers made. Since 100 rolls per hour are typical, the results represent roughly 3, 30, and 300 hours of craps.

The accompanying tables show how many of the 5,000 simulated solid citizens finished in various ranges, assuming they all had sufficient funds to outride whatever bad luck came in the interim. In each case, averages are close to the \$150, \$1,500, and \$15,000 predicted at \$0.50 per roll for 300, 3,000, and 30,000 rolls, respectively. The data show that with either bet, although numbers of players peak near these expected values, the spreads are relatively large. Further, as anticipated by the differences in volatility and skew, the ranges of wins and losses are wider betting \$82 four or 10 than \$41 each four and 10.

Simulation results also indicate how greatly volatility and skew overwhelm edge for statistically short sessions. For instance, after 300 rolls, many more bettors were \$400 or \$800 ahead than behind by the same amounts. But after 3,000 and more markedly 30,000 rolls, edge came to dominate and this situation reversed. Still, even after 3,000 rolls, a sizeable contingent was beating the casino. When 30,000 rolls had ensued, though, 4,971 of the four-or-10 bettors were losers while 4,996 of the four-and-10 crowd were in the soup. There are still spreads in the amounts, but they involve the degree rather than the fact of loss.

All demonstrating that while expectation shouldn't be taken lightly, it's hardly the be-all and end-all of gambling. The comparison of these two bets shows you can learn a lot about volatility and skew from Lyle and Lydia. Which, with a cue from the punters' poet, Sumner A Ingmark, recalls these lyrics written for the Marx Brother's movie, "At the Circus," by E Y Harburg:

Oh Lydia, oh Lydia, say, have you met Lydia?
She has eyes that folks adore so, and a torso even more so.
Lydia, oh Lydia, that encyclopidia.
Oh Lydia, The Queen of Tattoo.
On her back is The Battle of Waterloo.
Beside it, The Wreck of the Hesperus too.
And proudly above waves the red, white, and blue.
You can learn a lot from Lydia!