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Best of Alan Krigman
Playing It Smart: Why try for a small jackpot, with a bigger prize down the aisle?18 March 2009
Slot machines of the same denomination may have greatly disparate jackpots. A 3-coin-maximum quarter game might offer as little as $500, or as much as $10,000, $25,000, or more, for a $0.75 bet. A nearby dollar device could have less of a biggie, or something much more grand at $3 a pop.
You see solid citizens attacking them all. Maybe you wonder who'd shoot for a measly $500 instead of gunning for $50,000. Don't the bozos pay attention? Or is it enough easier to win a jackpot of $500 than $50,000, so why not go for a good shot at a day's pay rather than the extremely remote possibility of the mother lode?
Picture two hypothetical machines, each with 90 percent return edge or house advantage is 10 percent. Say the payoffs and their probabilities of occurrence are as shown in the following table. Payoffs and probabilities of occurrence for hypothetical 90 percent machines.
machine 1 machine 2 payoff probability payoff probability 500 0.001371% 10,000 0.000100% 75 0.013710% 500 0.002656% 50 0.102827% 50 0.019918% 20 0.514133% 20 0.099591% 10 2.056534% 10 0.398265% 5 6.169602% 5 9.501178% push 12.590792% push 23.174124% lose 78.551131% lose 66.804168%
In the conjectural machines, the chance of the 500-to-1 jackpot is indeed better than that of its 10,000-to-1 counterpart. But it's not exactly a lock. Stated in terms of odds, which are more readily envisioned than fractional percentages, the former comes in at 72,937-to-1 and the latter at 999,999-to-1.
Of more practical interest is the potential for the dough you bring to the casino being enough for a session which, if not profitable, at least won't knock you out of the box too quickly. For the sake of argument, Make believe you're anticipating about two hours of action. This might typically involve 1,200 spins.
With a bankroll of 100 times your bet, simulation shows the low-jackpot machine 1 gives you slightly under 50 percent chance of still being in the game, hopefully with a gain but if not with money left, at 1,200 spins. On the high-jackpot machine 2, your chance of making it is only approximately 1 percent.
Your level of optimism justifiably improves with a bankroll 200 times your bet, but not by much. The chance is 54 percent on machine 1 and 1.5 percent on machine 2. Staking yourself at 500 times your bet, the likelihood of survival jumps to 75 percent on machine 1 and 62 percent on machine 2.
Edge influences these phenomena. At $1 per spin, 1,200 rounds costs you a theoretical $120. The fact you can survive at all on a $100 stake, or bust out with $200 or $500 is a result of two other factors. Of these, the more familiar (not to imply well-known) is volatility picture it as the average change in your bankroll on every decision. The second of these characteristics is skewness an indication of the degree, if any, to which results favor small numbers of large wins or losses, or large numbers of small wins or losses, or something in between.
As for volatility, machine 1 has a standard deviation rating of 3.6 units per coup; the value for machine 2 is 10.5. Measures of skewness are even further apart. All traditional slots have high frequencies of small losses. On the win side, machine 1 in this example has reasonably numerous pushes and small returns, and fair instances of moderate scores; skewness is about 41. Machine 2 has moderate numbers of pushes and small wins, and few major hits; skewness is around 850.
Understanding edge, volatility, and skewness can help enquiring minds fathom the whys and wherefores of gambling. For everyone else, those who think it's all either luck or buttons being pushed by the bosses, just realizing how jackpot size anticipates prospects of going bust during a gaming session could be a boon. The beloved bard, Sumner A Ingmark, said it like this:
It isn't wise to pick games idly.
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