The sum of the parts ...

I'd like to try a little experiment to start this week's article. Suppose I give you the chance to select one of three indistinguishable envelopes. The choice of which one to pick is entirely up to you. Two of those envelopes are empty, but the third one contains a check for -- cue Dr. Evil -- $1 million! Next, I'll open one of the envelopes you didn't select to reveal that it does not contain the prize. Now, here's the million-dollar question. I'll offer you the opportunity to switch your envelope for the other unopened envelope. What do you do, hotshot? What do you do?

Ponder that for a bit while I give you another hypothetical situation, this time from the familiar world of fantasy baseball. We'll hold a minidraft and select two-man teams from the following four choices. Each day you can start only one of your pair, and unless both are off, you have to start one. Now, to be fair, because I've set the ground rules, I'll let you select the two you want first. Whom do you select?

I think it's pretty obvious that most people, if not everyone, would select Morneau and Pujols. In fact, I would question your sanity if you didn't. After all, there's no way I could beat your combination with Barton and Overbay in my incredibly meek arsenal, right? Take a look at the combined stats:

No contest, right? Here's where it gets a bit interesting, though. If you're one of those players who like to play the matchup game -- making switches in and out of your lineup daily in the hopes of always getting the best results from your starting lineup -- there's a chance that you could lose our little competition. Take a look at what was possible, given the rules of our competition, if I guessed right every day for Barton/Overbay and you guessed wrong each time for Morneau/Pujols:

Somehow, even though it seemingly defies all logic, a player who made all the right calls could have stolen victory from the jaws of defeat and outperformed two of the best hitters in the game with a pair of bats not considered worthy to be rostered in even one in 10 fantasy leagues. And therein lies the cautionary tale for today: Don't outthink yourself.

Much was discussed last week about the unpredictability of streaks and knowing exactly the right time to switch horses midstream. This week, we show you how you could be your own worst enemy by trying to squeeze too much out of your available talent.

Now, is it likely that you'd "guess incorrectly" every day while another owner would guess correctly? No; we're using the benefit of already knowing the results to demonstrate the extreme case. However, the idea that an owner would be far better off simply taking either Morneau or Pujols and riding him, day in and day out, is quite apparent after seeing the potential for disaster that exists. And if this kind of drop-off in performance can occur with these superstars, imagine the risks you take by attempting the same daily roll of the dice with players of lesser abilities! Surely every so often you might strike gold, as we did with Barton and Overbay, but the odds are stacked against you.

Incidentally, since Tuesday, the best of Barton/Overbay has continued to outshine that of their All-Star counterparts, winning the batting-average battle of the past week .417 to .185, scoring five more runs and winning the RBI battle 4 to 3. What does this prove? Nothing at all, really. It's just one possible outcome formed with the benefit of perfect hindsight. If you want to take a chance once in a while with sitting a stud against an opposing ace, there's no harm in that. However, if you're going to try to play the hot hand or better matchup regularly, you're likely cutting off your nose to spite your face. As much as common sense and intuition would have you believe that you have absolutely nothing to fear in taking such chances, the possibility of doing yourself more harm than good is very real. Why risk it?

Additionally, what exactly are you basing your "hunch" on when deciding to say, sit Miguel Cabrera for Casey Kotchman on a random day of the week? Is it the fact that Kotchman is 3-for-6 against that day's starter? Is it because the pitcher who is facing the Detroit Tigers has a career ERA of 1.03 against them for his career? If you want to find a reason to make this swap, these kinds of trends and statistical anomalies can always be found to justify the move. Sometimes it will work for you, but often it won't. Yes, there are cases -- I'm looking at you, Aramis Ramirez -- when you'll want to bench a highly drafted player until he gets his act together. Still, it just makes more sense, over the course of a full season, to stick with your original evaluation of a player and start your studs day in and day out.

Now, back to our initial experiment of the envelopes. Have you decided to switch, or does it make absolutely no difference? That's the most common answer to this mathematical poser, which is more commonly known as the Monty Hall problem. Most people will assume that once one of the envelopes is opened to reveal nothing, they now have a 50-50 chance to win the prize. Therefore, they would think, it doesn't matter whether you switch. Is that what you guessed?

Sorry to break the news to you, but that's incorrect.

When you first picked the envelope, you had a 1-in-3 chance to win the prize. That means there was a 2-in-3 chance that the check was in an envelope you didn't pick. Although I opened up an envelope that I knew did not contain the check, the initial odds of the exercise did not change. In other words, there was still a 2-in-3 chance that the check would be in an envelope you did not pick, and there would be only one envelope that would fit that description. Believe it or not, by switching envelopes, you actually would double your probability of winning from 1-in-3 to 2-in-3.

That might be counterintuitive, but it is mathematically the correct answer. But if you don't believe me, that's fine. How about you trade me Morneau for Barton and we'll call it even? What do you say? If you're so sure of yourself, let's make a deal!

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.