Estimating How Often You Should Win Is Different than Doing It.

Casino gamblers have wide latitude in tailoring their chances of winning or losing specific sessions. Within bounds imposed by the edge inherent in a game, this is achieved chiefly by sizing bets relative to a bankroll, raising or lowering wagers randomly or according to prior results, and setting gain and loss levels at which to quit. Choices essentially boil down to a small prospect of a big score, a good shot at a small win, or something intermediate -- offset by the complementary chance of a wipe-out.

There's neither magic nor mystery in such maneuvering. Picture two punters willing to dig as deeply as $200 into their pockets. The first has a double-or-nothing mindset -- and by playing skillfully can expect roughly four $400 wins for every eight $200 losses. The second is content with a 50 percent profit -- and can anticipate about eight $100 wins for every four $200 losses.

This sort of thing works because house advantage, the vehicle that drives casino earnings, only becomes evident over the long run -- thousands of patrons and millions of decisions. It doesn't actually draw a drop from the vein on every round. Over the short span characterizing an individual bettor in a particular session or during a specific day or weekend, fluctuations endemic to each game swamp the edge. Think about a single bet, say $1 in a slot machine with a 95 percent return. In theory, the house earns $0.05 per spin. In practice, nobody starts a round with $1 and ends it with $0.95. Instead, folks either lose $1, push by "winning" back the dollar, or hear the bells and whistles and earn anything from $1 to a lottery-class set-for-life jackpot.

Few casino aficionados indulge enough for probabilities entering the casino to reliably anticipate frequencies leaving, even over what seem like extended periods. So someone whose goal is to double $200 may have roughly one chance in three of success, but actually win more or fewer than four out of the dozen sessions experienced during a week, month, or year. And the solid citizen seeking $100 return on a $200 stake may be a 2-to-1 favorite, yet prevail on under or over the predicted eight out of a particular twelve attempts. Statisticians can be confident that after 12 million trials, gamblers would be within round-off error of 4 million wins for the first approach and 8 million for the second. But what do they have to say about the dozen casino visits?

To answer this question, the numero-noodniks root around in their bags of mathematical gee-gaws and pull out the old, reliable, handy-dandy "Poisson Distribution." This lets them estimate the probabilities, shown on the following list, for achieving from zero through 12 winning finishes in a dozen starts, when "risk of ruin" analyses give four or eight as the expected numbers.

Probability of achieving indicated number of winning sessions in 12 visits when expectation is four or eight wins
Number of successes	probability, four wins expected	probability, eight wins expected
0	1.83%	0.03%
1	7.33%	0.27%
2	14.65%	1.07%
3	19.54%	2.86%
4	19.54%	5.73%
5	15.63%	9.16%
6	10.42%	12.21%
7	5.95%	13.96%
8	2.98%	13.96%
9	1.32%	12.41%
10	0.53%	9.93%
11	0.19%	7.22%
12	0.06%	4.81%

Here are two examples of how to interpret the data for 12 visits.

1) When expectation is for four wins, players have 10.42 percent chance of six.

2) When eight wins are expected, players have 9.16 percent chance of only five. Adding the probabilities for successes below or above the projected values gives chances of falling short of or exceeding the mark. Bettors putting up $200 to win $400, who expect four victories, have over 43 percent chance of winning less often and 37 percent of more. Those risking $200 for $100 returns, who expect eight triumphs, have about 45 percent chance of fewer wins and 34 percent of more.

Admittedly, even knowing all this, I'm still surprised at stretches when I do worse or better than predicted. The poet, Sumner A Ingmark, prophesied this puzzlement when he penned:

With element of chance neglected,
False hope is pinned on what's expected.