House advantage or edge in casino games is related to the probability solid citizens will win or lose a round. The specific values are occasionally but not usually equal, however.

To illustrate, edge on any bet at single-zero roulette is 2.70 percent. But chance of winning depends on the particular wager. For instance, it's indeed 2.70 percent – a one out of 37 longshot – on single spots. Prospects are also long, four out of 37 or 10.81 percent, on four-number corners. They're nearly even, 18 out of 37 or 48.65 percent, on outside bets such as Red. And players have more chance of winning than losing, 24 out of 37 or 64.86 percent, on money split between two 12-number "dozens."

The key to reconciling edge and chance is the difference between the likelihood of winning and losing. In part, this accounts for pushes where no money changes hands. More significantly, it offers a way to weigh-in payoff. This is conveniently pictured in terms of "expected profit" – the chance of winning multiplied by payoff, and of "expected loss" – the probability of losing times the amount at risk. Taking the profit minus the loss components gives the net "expected value" of the wager. And edge is the fraction of the original wager represented by the expected value.

Pretend you bet $10 on Red at single-zero roulette. Payoff is even-money, $10. The expected profit is (18/37)x$10 = $180/37, $4.86. This bet can't push, so the chance of losing is 19/37. Expected loss is accordingly (19/37)x$10 = $190/37, $5.13. The difference, the expected value, is $4.86 - $5.13 = $0.27. You could also write this as $180/37 - $190/37 = ¬-$10/37 or -$0.27. And 27 cents is 2.7 percent of the $10 bet. The minus sign indicates that the 2.7 percent edge favors the house.

A $10 bet on the corner of four numbers pays $80. Expected profit is (4/37)x$80 =$320/37; expected loss is (33/37)x$10 = $330/37. The expected value of the bet is accordingly $320/37 - $330/27 = -$10/37 or -$0.27. Divide by the $10 bet to find that edge is still 2.7 percent. By going long, for 8-to-1 rather than even money, you've cut your chance of success but haven't altered the edge enjoyed by the casino. In the extreme for this game, $10 up for grabs on a single spot, chance of winning is a remote one out of 37 to win $350. Expected profit minus expected loss is (1/37)x$350 - (36/37)x$10 = $350/37 - $360/37 or, again, -$10/37.

This works in the other direction, as well. Make believe you bet a total of $10 by dropping $5 on each of two "dozens" at a single-zero table. Individually, each dozen would pay 2-to-1 or $10. You can't win them both simultaneously. Effectively, therefore, you're looking to pick up $10 on one of the dozens and lose $5 on the second, for a net of $5. The downside is to lose the whole $10 on zero or the third dozen. The expected value is (24/37)x$5 - (13/37)x$10 = $120/37 - $130/27 or the familiar -$10/37, -$0.27, 2.7 percent of your bet. Edge hasn't changed, but you have a much greater chance of winning than losing.

It all comes out in the wash for the bosses. As they get more and more action, the central limit theorem of probability predicts they move closer and closer to the values predicted by the math. One high roller betting $1 million on Red may win or lose $1 million. A wonderful human being (just ask any host!) betting a million on a corner may win $8 million or lose $1 million. And, a wealthy individual splitting $1 million across two dozens may win $500,000 or lose $1 million. But, say, 100,000 of the hoi polloi bet $10 each, for a combined total of $1 million. On Red, some of these wagers will pay players $10 and others will cost them $10. On the corner, some will pay $80 and others will cost $10. And, on the dozens, some will win $5 and others will lose $10. But the proportions, in the end, will be such that the casino keeps a sum statistically close to 2.7 percent of $1 million – $27,000.

One betting strategy may give you more pleasure than another. None will circumvent the laws of the known universe. Sumner A Ingmark, the math mavens' revered rhyme-writer, put it like this:

 

A business that keeps up the store,
Knows two plus two add up to four,
And neither less nor one bit more.