How does the house edge work?

The house edge remains a mysterious thing to casino players, including the one who asked me recently about roulette.

"Does a 5.26 percent house edge mean the house wins 52.6 percent of all spins?"

That wouldn't be a bad guess if every wager paid even money, and the house edge was made up entirely of the difference between the frequency of house wins and the frequency of player wins. In the red-black wager at roulette, where winners are paid even money, the house wins 52.63 percent of rolls, the player wins 47.37 percent. Subtract 47.37 from 52.63 and you get the 5.26 percent house edge.

It is the difference between win percentage and loss percentage that's important, and you can't just multiply the house edge by 10 and get the percentage of losing wagers. Roulette is a bit of a coincidence that way.

Take craps and the pass line wager, another even-money payoff. The house has a 1.41 percent edge.
Obviously, it wins more than 14.1 percent of the time. Instead, if you want to know the frequency of house wins, divide that 1.41 percent house edge in half, then add the result to 50 percent. The house wins 50.705 percent of wagers, the player wins 49.295 percent. Do the basic subtraction, and you get a difference of 1.41 percent --- the house edge.

Things get more complicated when payoffs get bigger. Let's go back to roulette and single-number bets. A double-zero roulette wheel has 38 numbers, with 1 through 36 as well as 0 and 00. Any time you bet on a single number, you have one way to win, and 37 ways to loses.

Yet the house edge is the same 5.26 percent as it is on red-black, odd-even or first 18-last 18. On those wagers with even-money payoffs, the house wins 52.63 percent of the time, as we have already seen.

Obviously, the house wins far more than 52.63 percent of single-number wagers. It wins 37 of 38, or 97.37 percent of wheel spins.

The reason you're bucking a house edge of 5.26 percent even though you lose 97.37 percent of the time comes in the payoff, of course. Winning single-number bets are paid at 35-1 odds. If it was a truly even bet, you'd be paid 37-1.

The difference between the payoff and the true odds is the key to the house edge. Here's the way it works.

Let's say you bet $10 on 17 on each spin of a perfect sequence in which each number turns up once. You risk a total of $380. On your one win, you get back $360 --- $350 in winnings for the 35-1 payoff, plus the return of your $10 wager on that spin.

That means the house has kept $20 of your $380 in wagers. Divide 20 by 380, and you get .0526. Multiply by that by 100 to get percent, and you see the house has kept 5.26 percent of the money you've wagered.

That's the house edge on single number bets.

It's simple enough to do the same for place bets or proposition wagers in craps, or the player bet in baccarat, or spaces on the Big Six wheel. The house edge is not based on the frequency of winning hands alone, nor is it based solely on the payoffs on winning wagers. Nor is a game with frequent winners necessarily one with a low house edge, nor a game with low payoffs per win a high house-edge proposition.

From a player's perspective, it's not how often you win, nor how much that counts, it's how much how often, working together.

Do casinos make their profit from winning players?

Q. Sitting in the buffet, I caught a snippet of conversation that I was wondering if you could explain. (I wasn't listening in, they were so loud I couldn't help hearing.) One guy was saying that the casino really makes its money off the winners, and the other guy said something like, "Oh really? You mean we pay for all this when we win?" That doesn't really make sense to me.

A. Kind of leaves you wondering what kind of gambling palaces they could build if everybody won, doesn't it? If the casino makes money off the winners, then more winners must mean more profits, right?

But seriously, there is a way of looking at how the casino makes its money that looks at casino profits as a tax on the winners. Casino games make money because they pay the winners at less than true odds. If 38 people are sitting at a double-zero roulette table and each bet $1 on a different number on a single spin, the 37 losers each will lose their buck, and the one winner will be paid at 35-1 odds and walk away with $36. If the casino was paying true odds, the one winner should be paid at 37-1 odds and walk away with $38. The casino profit is the $2 not paid to the winner that he'd get if true odds were paid.

Same deal with sports betting. In most sports books, you have to put down 10 percent vigorish on top of your bet. Let's say you and I are betting on the same football game, with me betting on Team A and you on Team B. We each intend to bet $100, but we have to pay the vig, so we actually each bet $110. When my team wins --- hey, it's my example; I get to win --- you lose your $110 bet, but I'm paid only $100, along with the return of my $110 wager. The casino profit is the $10 it didn't pay me on my winning bet.

Sometimes the paying of winners at less than true odds is disguised a bit. In baccarat, for instance, bets on banker win more often than they lose, and bets seem to be paid at even money. However, bettors have to pay a 5 percent commission on winning bets, so winners aren't really paid at 1-1; they're paid at (1 minus .05)-1, and that's less than the true odds of winning the wager.

So it goes with every casino game. There are going to be winners, and there are going to be losers, but the house will make money because it pays winners less than the true odds of winning the bet.

Time, money and the house edge

If you've been my casinos columns for very long, you've seen the numbers a hundred times: The house has a 5.26 percent edge on American double-zero roulette, a 1.41 percent edge on the pass line at craps, a 2 to 2.5 percent edge against the average blackjack player, but only a half-percent or so edge against a basic strategy player.

Chances are you use those numbers as a rough comparison of games. Which give you the better shot to win? Which are nothing more than "house" games? And that's fine. You do have a much better shot to win if you're betting the pass line at craps than if you're playing roulette.

But those numbers are also statistical averages. They don't mean that every time you bet $100 on the pass line, you're going to lose $1.41, or even that every 100 times you bet $100 on pass, you'll lose $141.

If the losses were that regular, and that certain in the short term, no one would play. Even on the casino games with the highest house edges, results are volatile enough that players will win sometimes.

Given enough trials, though, the house edge will hold up, and the more trials there are, the closer the results will be to giving the casino its expected profit.

One of the more colorful descriptions of how this all works came from the late Peter Griffin, author of the classic The Theory of Blackjack. I'd been invited to spend a week as a student at the Harrah's Institute for Casino Entertainment in Las Vegas, a crash course in casino operations designed to give gaming basics to non-gaming employees Harrah's had marked as up-and-comers. Griffin was a guest instructor.

Imagine a situation, he told us, in which a busload of roulette players have finished for the day and are in the waiting room, waiting to board for the trip home. They decide they have time for one last spin of the wheel if each of the 100 players gives $100 to a casino manager with instructions to place the bets. (This takes place in a jurisdiction with a particularly lenient gaming board, I might add.)

The manager puts all the money --- $10,000 worth --- in a big box, and climbs a staircase to a balcony overlooking the waiting room. He then takes out $526 --- more or less --- and dumps the rest of the money back over the railing to the waiting bus group.

Players scramble for the money. Some get back their $100. Most get back less. Some get more --- a few get considerably more.

There are winners. There are losers. And the house has its 5.26 percent.

Slot machines work the same way. When we say quarter slots at a given casino return an average of 93 percent to players, that's just another way of saying the house edge on those machines is 7 percent. If you play $100 through a quarter slot at that casino, are you likely to wind up with precisely $93? No, most of the time you'll wind up with less. Much of a slot machine's long-term return is tied up in larger jackpots, so sometimes you'll win hundreds, or even thousands of dollars for that $100 in play. You may win big, or more often lose fast, but on balance over hundreds of thousands of plays, the house will get its 7 percent.

If you go to the casino and play a couple of hours on a three-reel quarter slot, you might spin the reels 1,000 times --- 500 plays per hour is a busy, but not hectic, pace. With a three-coin maximum bet, you're risking $750. If the machines in that casino return 93 percent, your expected average loss is $52.50, but you won't land precisely on that figure very often. Most of the time, you'll lose somewhat more than that. Less often, you'll get a big hit or a few medium-sized wins and walk away a winner. In one short trip to the casino, there's no telling what your results will be.

But let's take a page out of Griffin's book and say you're on a bus trip with 100 players all spending a couple of hours on those 93-percent, quarter slots, all betting 75 cents a spin for a total of 1,000 spins each. Now your group is setting the reels spinning a total of 100,000 times, and the casino can almost count on getting something close to its 7 percent take.

There almost certainly will be a winner, or several winners, in your group. There also almost certainly will be those who lose fast and wind up dropping a couple of hundred dollars. The majority will lose some of their money. All told, your group is likely to leave behind a total very close to $5,250 on its $75,000 worth of wagers.

There are winners. There are losers. And the house has its 7 percent.

That's the way it is when you look around any busy casino. A few are winning. Most are paying for their day's entertainment. It's the hope that this time we'll be one of the winners that keeps us going.

Does the house make its money off winners?

Q. Sitting in the buffet, I caught a snippet of conversation that I was wondering if you could explain. (I wasn't listening in, they were so loud I couldn't help hearing.) One guy was saying that the casino really makes its money off the winners, and the other guy said something like, "Oh really? You mean we pay for all this when we win?" That doesn't really make sense to me.

A. Kind of leaves you wondering what kind of gambling palaces they could build if everybody won, doesn't it? If the casino makes money off the winners, then more winners must mean more profits, right?

But seriously, there is a way of looking at how the casino makes its money that looks at casino profits as a tax on the winners. Casino games make money because they pay the winners at less than true odds. If 38 people are sitting at a double-zero roulette table and each bet $1 on a different number on a single spin, the 37 losers each will lose their buck, and the one winner will be paid at 35:1 odds and walk away with $36. If the casino was paying true odds, the one winner should be paid at 37:1 odds and walk away with $38. The casino profit is the $2 not paid to the winner that he'd get if true odds were paid.

Same deal with sports betting. In most sports books, you have to put down 10 percent vigorish on top of your bet. Let's say you and I are betting on the same football game, with me betting on Team A and you on Team B. We each intend to bet $100, but we have to pay the vigorish, so we actually each bet $110. When my team wins --- hey, it's my example; I get to win --- you lose your $110 bet, but I'm paid only $100, along with the return of my $110 wager. The casino profit is the $10 it didn't pay me on my winning bet.

Sometimes the paying of winners at less than true odds is disguised a bit. In baccarat, for instance, bets on banker win more often than they lose, and bets seem to be paid at even money. However, bettors have to pay a 5 percent commission on winning bets, so winners aren't really paid at 1:1; they're paid at (1 minus .05):1, and that's less than the true odds of winning the wager.

So it goes with every casino game. There are going to be winners, and there are going to be losers, but the house will make money because it pays winners less than the true odds of winning the bet.