Playing It Smart: Why most casino gamblers finish sessions below average

Casinos profit on the advantage they have on every bet. This advantage is generally expressed as an edge, expected value, or return (sometimes called payback) percentage. Edge is the fraction of a bet the casino expects to earn, averaged over many trials. Expected value and return or payback percentage are the average fractions of bets players recoup after the edge is deducted. So, if edge is 5 percent (the house earns an average of $0.05 per dollar bet), expected value and return percentage are 95 percent (the house pays out an average of $0.95 per dollar).

These values are crucial to the casinos because they indicate the profit margins on the action they book. Solid citizens often use them to judge the quality of a gamble. High edge -- low expected value, bad; low edge, high expected value, good. However, these measures shouldn't be interpreted as predictors of the outcomes of individual coups or games. Certainly not exactly, or nobody would gamble. But, not even as a guide to a reasonable range.

An admittedly extreme hypothetical example shows one aspect of why this is so. Picture a $1 longshot with odds of 9,999-to-1. Winners get $9,999 plus their original dollar. On the average, per 10,000 players, 9,999 lose $1 each and one wins $9,999.

The casino washes out so edge is zero and the expected value of a dollar bet is $1. The vast majority of players lose $1, while a tiny minority win $9,999. What if the odds stayed the same but the payoff was $7,499? The house would have 25 percent edge and the expected value of a dollar bet would be $0.75. The masses would still lose $1 while the elect would win $7,499.

Nobody leaves with the expected value in either case. Further, hardly anybody would decide whether or not to take a flier based on the expected value of a dollar bet being $1 or $0.75. Players are more apt to think about the most likely value of their bets being zero, by overwhelming odds of 9,999-to-1. The question is whether it's worth tossing away a buck on the happenstance they'll luck out and cop enough for a giant flat-panel HD TV set.

A blackjack session offers a more realistic situation. With perfect Basic Strategy, edge is about half a percent. Pretend you play for three hours at a table with four other people, each at one spot. You'll get in about 260 rounds. At a flat $10 per hand, your gross wager will be $2,600 the edge on which would amount to $13. Had you begun with $500, strict interpretation of the laws of probability puts the expected value of your $500 stake at $487. If this is where you finish, it will be pure coincidence.

Because of volatility, the customary application of the laws of probability say you should be 95 percent confident of ending within $357 of the expected value up or down. That is, between $130 and $844. But, owing to biases applicable to individuals, players are far more apt to be broke or way above $844.

One such bias arises because expected value calculations assume bankrolls are unlimited. That is, players can keep betting with the possibility of recovering. In fact, once a gambler hits bottom, there's no money left to continue. Many more players therefore go home with zero than the math mavens prophesy.

A second bias results from many players regarding casino gambling as an opportunity to risk money they can easily get, on a chance at an otherwise impossible dream. These folks don't quit after a round leaves them near the expected value with a small loss or a modest profit. Rather, they persist until they go belly-up or fulfill their fantasies. The casinos doesn't care because their averages after lots of action will still be close to the expected value. But for the bulk of individual bettors, the most likely final value of a bankroll is far below the average or expected value often zero. This is offset by sporadic players high above average and a conservative punter or two in between.

All of which helps explain why the beloved bard, Sumner A Ingmark, was maudlinly motivated to muse: