How Realistic Are Your Casino Profit Goals?

What's the chance of entering a casino with one pile of pelf and leaving with some greater amount, assuming you go for broke? That is, growing stake "S" to fortune "F" before busting out. In the limit, a "fair" game with no edge, the probability equals S divided by F. Bet size doesn't enter into this "ideal" equation. Nor does volatility, as determined by factors like payoff ratios. Both, however, affect how long it might take to do or die.

Make believe the numbers are $100 and $5,000. The answer is then 100/5,000, the same as one out of 50, or 2 percent. This is easy to understand intuitively. Say that a plethora of players each takes a shot. With no edge, overall results should be a wash. Therefore, for every happy camper who wins $4,900 and finishes with $5,000, 49 lose their $100 bankrolls. That's one out of 50.

Here are other idealized examples. Risk $250, double-or-nothing, to amass $500; the chance is 250/500, one out of two or 50 percent. Load a machine with $25 and keep playing until it's history or you hit a jackpot and have $25,000; the probability is 25/25,000, one out of a thousand or 0.1 percent. How about buying in with $1,000 and trying to eke out a modest $250 profit? The likelihood of success is 1,000/1,250, an impressive 80 percent.

Of course, this is just theoretical because casinos don't offer fair games with no edge. House advantage muddies the waters. But, in practice, its effect on your potential for triumph is minor.

Consider starting with $10 on Pass at craps, taking no odds, and parlaying profits until you either go broke or win $1,280 and walk with $1,290. Pass has a 1.4 percent edge and a 49.29 percent probability of winning. Ideally, your ultimate chance is 10/1,290 or 0.775 percent. To actually achieve this target you'd have to have a run of seven wins. The probability of this feat is 49.29 percent multiplied by itself seven times, or 0.707 percent.

Assume instead your parlay is on a much less favorable even-money $10 bet. For instance, Red at double-zero roulette, on a table where you lose your entire wager on Black, 0, or 00. The edge here is 5.26 percent and the chance of winning is 18/38 or 47.37 percent. The ideal probability of getting from $10 to $1,290 is still 10/1,290 or 0.775 percent. Winning seven bets on Red in a row has a true probability of 18/36 multiplied by itself seven times, or 0.535 percent. The greater edge trims the outlook more than for bets on Pass. But the difference is still not excessive.

Think about the same roulette table but bet on a double-row, six numbers in all, paying 5-to-1. Edge is still 5.26 percent but volatility is higher. Start with $5 and parlay wins in quest of $1,080 profit. The ideal probability is 5/1,085 or 0.461 percent. In reality, you need three wins in a row. The chance is 6/38 multiplied by itself three times or 0.395 percent. Less than the ideal but closer, comparatively, than it was for the bet on Red.

You can go further, using fancy "risk of ruin" calculations, to ascertain the influence of bet size. (Relax, I'll just give you some results, not bore you with complicated formulas.)

Pretend you want to go from a stake of $100 to a final fortune of $1,000 betting on the pass line at craps with no odds. The ideal probability would be 100/1,000 or 10 percent. Risking $10 per round, your actual chance would be 2 percent. Pump your wager to $25 per round and your prospects jump to 5.7 percent. At $50 per round, you're up to 7.7 percent. And with $100 on the line, you have an 8.8 percent chance of success.

Dividing stake by desired final fortune yields a good, albeit optimistic, educated guess as to the likelihood you'll soar rather than crash when you gamble. House advantage cuts chances relative to the ideal -- the impact falling as edge shrinks, bets increase, and volatility rises. But, few solid citizens have a clue as to the chances of reaching their goals, anyway. So using the ideal estimate is a giant step forward. As the inestimable epigraphist, Sumner A Ingmark, elegantly engrossed:

Quite often calculations rough,
Yield answers more than good enough.