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Best of Alan Krigman

Gaming Guru

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How Skewness in Gambling Affects Your Session Hopes and Fears

18 January 2006

To casinos, edge should be the only intrinsic statistical parameter of games that matters. Because, with enough bets of roughly comparable amounts over extended periods, averages prevail and the bosses end up winning close to the percentage of the gross wager the math predicts players will lose.

But individual patrons rarely play enough for the law of averages to rule their action. For such solid citizens, parameters in addition to edge may strongly influence short-term results.

The volatility of a game, for example, anticipates how much a bankroll tends to rise or fall from round to round as well as during a session or casino visit. Volatility is measured by what the math mavens call "standard deviation." Games with equal edge may differ in volatility. And higher standard deviations generally mean more extreme bankroll swings. These may help put betters both over the top or onto the sidelines. But, which?

Standard deviation can't tell up from down. A 40 percent chance to win $15 on a $10 bet, plus $15 or minus $10 with bad news half again as likely as good, has a standard deviation of $12.25. Useful for predicting the size but not the direction of swings.

Skewness, or skew, may yield a clue here. Zero skew implies symmetrical gains and losses. Negative skew indicates lots of minor wins offset by rare major disappointments. And positive skew suggests many small setbacks and occasional rich returns.

Hypothetical games can be devised having the same negative expectation and standard deviation but moderately different skews. These afford ability to isolate skew from the other parameters. And, such games can then be simulated on a computer to study the impact of skew on factors important to players such as reaching typical low and high points in archetypal sessions.

Consider three such imaginary games with the following characteristics: a) bet $0.32 on a 3.6 percent chance to win $6.78, b) bet $1.39 on a 50 percent chance to win $1.25, and c) bet $6.92 on a 96 percent chance to win $0.19. In all three, loss due to edge is 6.7 cents and standard deviation is $1.32 per round. Skewness is +5, zero, and 5, respectively. Incidentally, these figures aren't pulled entirely from a hat. Place bets on 10 at craps, proportioned to wagers of $1 to win $1.80, have 6.7 cents expected loss and $1.32 standard deviation with a skew of 0.71. Laying against the 10, betting $1.93 to win $0.87, has this expected loss and standard deviation with a skew of +0.71.

To get an idea of the effect of skew, the three games were simulated for sessions of up to 250 decisions. One run had loss limits of $50 and win goals of $40 like stakes of $500 and profit objectives of $400 on bets of $10 per round.

Despite the relatively high edge, few players bit the dust, although the moderately low volatility also kept most from hitting their win goals. With skew of +5, only 6.4 percent went to jail and 1.7 percent passed "go." With skew of zero, the figures were 8.6 percent to the lockers and 1.0 percent to the winners' circle. And with skew of -5, 8.9 percent were in the soup and 0.1 percent in the clover. Under the stated conditions, independently of volatility and expected loss due to edge, chances of failure were least and of success greatest with the positive skewness. Conversely with the negative skew.

Simulated sessions assuming deeper pockets and more modest aspirations showed a similar trend. The opposite objectives, risking smaller bankrolls to fulfill wilder fantasies saw the highest mortality and greatest triumphs together at the skew of +5, with the best survivals but the worst rates of victory at -5.

This should leave you wanting to know more about skew and how it abets or hinders the gambling trade-offs you make. And, of course, recognizing that while luck has its role, the poet Sumner A Ingmark was on the money when he rhymingly recommended:

Investigate properties that are inherent,
To hope for what's normal and not what's aberrant.
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.