Odd streaks, past results and future outcomes

For about a lifetime now, I've been telling gamblers that in most games, past results have no effect on future outcomes. (Blackjack is an exception, since each card dealt out changes the composition of the remaining deck, altering the odds of the game.)

The roulette ball has landed on eight black numbers in a row? The chances of the next one being black are still 18 in 38. The craps shooter has gone a dozen rolls without a 7? Unless you're dealing with a controlled roller, the chances of the next roll being a 7 are 1 in 6, same as always. You've just hit a big slot jackpot? The odds against the big jackpot combination turning up on the next spin are the same as the spin before.

From time to time, I'm asked how that possibly can be. If in the long run, 18 of every 38 spins of the roulette wheel will be red, and there have been eight black numbers in a row, shouldn't you be betting on red? If there have been a dozen craps rolls without a 7, isn't 7 "due"? After all, in the long run, one of every six rolls will be a 7.

The answer is that there doesn't have to be any makeup time. In any game, there will be unusual streaks that seem to defy the odds. Casinos and their customers count on that. Without such streaks, there would be no winners --- we'd all just be handing over our money at a prescribed percentage.

Eventually, such streaks just fade into statistical insignificance, overwhelmed by the sheer number of trials that go on in casinos.

For a simple example of how all this works, I like to use another chance event: flipping a coin. Just as in rolling the dice, spinning the roulette wheel or pulling the slot handle, past coin flips have no affect on future outcomes. There's a 50-50 chance of heads or tails on every flip, regardless of what has gone on before.

This came up recently on an Internet message board in which I participate in the odd discussion --- the odder, the better. The manager of a softball team said a coin flip determined who was the home team, and that he'd lost 13 consecutive flips. What were the odds?

On a 50-50 event like flipping a coin, calculating the odds is easy. Your chances of losing a single flip are 1 in 2. Your chances of losing 13 in a row are 1 in 2 to the 13th power --- start with 2, then multiply by 2 12 times. The answer is 1 in 8,192. In 8,192 sets of 13 coin flips, on the average you'll lose all 13 of them once.

Another poster on this message board suggested you could improve your odds with some judicious selections. "If it hits seven heads in a row, isn't it more likely to come up tails the next time, because of the law of averages?"

No, the odds never change. If heads come up seven times in a row, the odds on the next flip are still even. There never has to be a makeup period --- in the long run, any unusual streaks just fade into statistical insignificance.

"But in the long run, things even out," the poster said. "That leads me to believe that, since you know things will even out, you're more likely to hit the other side if there's been a run on one side."

Recognize that train of thought? That's the same thing as a roulette player who thinks that after eight black numbers, red is due, or a craps player who thinks that after a dozen rolls without a 7, the shooter must be due to roll a 7.

But any evening out to be done is purely statistical, whether we're talking roulette, craps, slots or coin flips.

Let's say the coin flipper starts with 7 heads in a row. At that point, 100 percent of flips have been heads. The expectation is that 50 percent should be heads, so it appears there's evening up to do.

Now let's say future flips come up exactly 50 percent heads and 50 percent tails. (Things rarely come out quite so neatly, of course.) After another 100 flips, 57 have been heads, and 50 tails. Now only 53.3 percent of the flips have been heads, only 3.3 percent above expected average. On thousand flips after the first seven, we have 507 heads and 500 tails. 50.3 percent have been heads. One million flips after the first 7, 500,007, or 50.000035 percent, have been heads.

There has been no "making up," period, but we're right on 50 percent heads and 50 percent tails. The original streak of seven heads has just faded against the statistical background.

Streaks happen, and if you're on the winning side, streaks are meant to savor. But there doesn't have to be any equal but opposite streak to make the odds come out right. Given enough trials, normal results from the streak onward are enough to keep the games on track.

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.