A look at the gambler's fallacy

Some of my regular readers may know that I used to be a dealer at an Atlantic City casino. One of the games I would be assigned to run on a semi-regular basis was roulette. For those of you who may not have ever frequented the gambling floor, next to each roulette table is a towering display that advertises the most recent winning numbers at the table. Inevitably, during the course of an eight-hour shift, at least one of the tables will suddenly fill up with players, lured by the siren song of a "streak."

"Look at that table! Ten reds in a row! Unbelievable!"

Yes, one of the bets on the table is a simple 50-50 proposition in which you bet on whether the next number the ball lands on will be either red or black. Now technically, it's not completely 50-50 because there are two green numbers -- 0 and 00 -- which give the house a slight edge on this wager, but there is an equal number of red and black numbers, so we'll go with 50-50 for the purposes of this discussion.

"Another red! Eleven in a row! I've never seen anything like it before!"

At this point, a very strange thing happens at the table. Half of the customers are convinced that this wheel is lucky for red, and that the next ball will surely land once again on a red number. The other half are equally convinced that such a streak cannot possibly hold up. Black, they argue, is "due" and besides, what are the odds that red would ever come up 12 times in a row? The truth is that neither group is correct. Each spin is a completely independent random event, and the odds of the next number to be red are still exactly the same as the odds of it to be black.

This lapse in logic is called the gambler's fallacy, and it's a big reason why those multimillion-dollar hotels remain in business. However, misconceptions about laws of averages aren't exclusive to the casino. Fantasy sports are chock full of examples of similar situations in which all of the math in the world simply won't help you sort out the mysteries of the future. If you want to know the exact location of Mars in the night sky on Sept. 1, 2010, mathematics can provide you with the precise answer, down to the millimeter, right now. Knowing what Martin Prado's batting average will be on that same day? Good luck getting within 15 points.

The Hot Streak

On April 17, Colby Rasmus of the St. Louis Cardinals went 0-for-3 against the New York Mets and his batting average sat at a dismal .179 for 2010. During the next 10 games, Rasmus went 16-for-33 with four home runs and nearly doubled his average to .344 overall. So, which fantasy advice for the next few games makes the most sense to you?

• Rasmus is swinging a hot bat. No way you should take him out of your lineup.
• There's no way Rasmus can keep this up. He's due for a cold spell. Get him out of your lineup.
• Rasmus is a career .262 hitter. That's what you should expect in the next two weeks.

The answer is that none of these arguments is logically sound. Each at-bat is an independent event, and while it is true that you'd expect someone with a history of batting .350 to perform better on a regular basis than a player who barely keeps his head above the Mendoza Line, for any given subset of games, anything can happen. Depending on the matchups, the weather, his health and a myriad of other factors, Rasmus is just as likely to continue his hot streak as he is to regress to his career average, or sink back into a slump of epic proportions.

The Bad Start

Let's look at notorious slow starter Mark Teixeira of the New York Yankees. A career .236 hitter in April, somehow Tex always seems to land on his feet by the end of the season, finishing in the neighborhood of .290 or so. In fact, our ESPN projections had proposed that Teixeira would finish this season hitting .299 and nobody seemed to think this was out of line. So, assuming the projection was a solid one, and given the fact that Teixeira was 11-for-79 (.139) on the morning of April 30, what statement makes the most sense to you?

• Teixeira was projected to be a .299 hitter, so forget the first month; he'll hit .299 the rest of the way.
• Teixeira will hit .323 the rest of the way so that by the end of the year, his average will indeed be .299.
• Teixeira has dug himself too big a hole this year. At best, he'll get back to .270 by season's end.

Again, while you can argue any of these mathematically, not one of these statements is any more or less likely to occur than any other. Batting averages, even those based on years of individual performance data, do not have a set outcome from a probability standpoint. Certainly, your expectations are colored by a player's past seasons, but declining/improving skills, different playing environments and roles, and sheer "luck" all make predicting batting averages too volatile a market for any true success.

Personal History

When it comes to the roulette wheel, there is no "memory." In other words, past spins have no impact on future results. Does that translate to fantasy baseball as well? Consider Roy Oswalt of the Houston Astros. Before Thursday's game, he was 23-1 lifetime against the Cincinnati Reds … and he lost, 3-2. What conclusion would you draw based on this?

• Oswalt's past success gave him no extra advantage against the Reds.
• The Reds were "due" to beat Oswalt. We should have expected the result.
• Now that the current Reds have beaten Oswalt, they'll probably beat him again the next time they play.

I would disagree with all three statements. Oswalt knew his record against the Reds, or at least had to have a pretty good idea that he usually seemed to pitch well against them. Much of baseball is mental, and if a player goes into a game relaxed and confident, he is probably going to perform better than if he goes in thinking he is doomed to failure. Having a positive history isn't a guarantee of future success, but it absolutely plays a part. Having said that, just because a team or player is confident -- say as the Reds might be the next time they face Oswalt -- that doesn't guarantee victory, either. If that were true, we'd never see any upsets. And as for being due, the record was 23-1. I'd say they've been due for several years now. That argument makes little sense.

Certainly, you do have to take personal histories with a grain of salt. After all, Jamie Moyer is 18-4 lifetime against the Baltimore Orioles. Since many of those wins came against a lineup that included Roberto Alomar, Cal Ripken Jr., Eddie Murray and Rafael Palmeiro, it's not exactly relevant to 2010 if he were somehow to go up against them. However, his 13-4 record against the Washington Nationals, including 7-0 since 2007, isn't something you should dismiss as completely irrelevant, even if you shouldn't automatically chalk one up in the win column for the Philadelphia Phillies, either.

The fact is that all statistics need to be taken in context. They can give us a general idea of how well players are doing in relation to one another, but they are only a snapshot of that particular moment in time. When it comes to predicting the future outcome of small slices of the season, past results can help color our opinions, but they remain just that -- opinions. It's better than using tarot cards or horoscopes as a guide, but in the end, they're just as fallible.

Remember, barring rainouts, it's a mathematical certainty that the average major league team will win 81 games in 2010. If you think you can figure out how many games the Los Angeles Dodgers are going to win the rest of the way using that fact as a starting point, then I've got a roulette wheel with your name on it.

AJ Mass is a fantasy baseball, football and college basketball analyst for ESPN.com. You can follow AJ on Twitter or e-mail him here.