In a recent article, I wrote about how "investing" in lottery tickets was a surefire way to lose money. So did Tom and Dave Gardner in this year's April Fool's Day joke. The chance of not winning the jackpot in, say, the Powerball is 99.99999917%, or as close to a sure-thing loss as one can find.

In buying lottery tickets, as with any other money transaction, there is a measure of risk analysis. Every time we part with our cash, there is a risk that we won't get something of value in return. When buying a lottery ticket, the risk of not receiving the jackpot is very high. When buying groceries, the risk of getting something bad is very low. When buying a movie ticket, the risk is slightly higher -- after all, the movie could be a dud.

When we buy stocks, though, the risk of not getting something valuable in return is somewhere between these extremes. You shell out money in a stock purchase, and that money is exchanged for part ownership in a company. This entails some risk. For instance, the company could go belly-up and you would end up losing your money. However, the gain is potentially higher than when buying food or entertainment. We could get cash paid to us in dividends or the stock price could go up and up.

As investors, we must perform some risk analysis. We must weigh the chance of a gain against the risk of a loss. One of the questions smart investors try to answer, therefore, is what is the risk that something bad will happen? Probability raises its head, but with the added twist of information.

Let's Make a Deal
Here is a favorite probability puzzle of mine that illustrates how information can change a situation. You may remember the old TV game show, Let's Make a Deal. In that show, Monty Hall offered contestants whatever was behind a curtain or a door, and then tried to persuade them to change their mind with usually amusing results, regardless of their choices. Sometimes the player got a car, sometimes they got a goat.

In this puzzle, Monty offers you whatever is behind one of three doors. He tells you that there is a car behind one of the doors -- he knows which one -- and a goat behind each of the other two. After you make your initial choice, he always opens one of the other two doors to display a goat. Then he asks if you want to keep your original choice or switch your choice to the other remaining door. What strategy gives you the best chance of winning?

The correct answer is to always switch. You will improve your odds of winning from 1-in-3 to 2-in-3.

Huh?

How can that be right? If, like many people, you think the second choice presents you with a 1-in-2 chance of picking the car -- from either door -- the answer seems counterintuitive.

What happens is that Monty has given you some information about the system. Remember, he knows where the car is and won't show it to you at this point in the game. When you make your original choice, you have a 1-in-3 chance of being right. Simple enough, so far.

That means two-thirds of the time, the car will be behind one of the two doors you did not pick. Think of them as a "group" -- and two-thirds of the time, as we just discussed, the car will be found within that group of two. Again, the odds are 1-in-3 for your door, 2-in-3 for the group. Given the chance outright to pick either one door or the group, you'd obviously go with the group, as 2-in-3 odds are better than 1-in-3 odds.

Now pretend I do two things: I show you one door within the group that doesn't contain the car -- remember, we just agreed this group of two doors collectively has 2-in-3 probability of containing the car -- and give you the chance to pick the group again, which now has its former 2-in-3 probability stuffed behind just one door.

In other words, in the second round you haven't been presented with a fresh 50-50 choice, but you've been gifted with the chance to bet on the one remaining representative of a collective unit that still has a 2-in-3 chance of containing the car.

Here is another way to think of it. The only way you can lose by switching is if you were correct in the first place. Since that "first place" was correct only one-third of the time, you would lose only one-third of the time by switching. Losing one-third of the time means that you win two-thirds of the time, if you switch.

Still don't believe me? Here's more on this.

Mr. Market and Monty Hall
In the stock market, we are given the choice to choose among many doors, and we can even choose more than one. But which ones to choose? At first, our choices may be random, based on tips, rumors, and talking heads. But one of the great things about the market is that there is information available for the diligent stock picker. Not only is Mr. Market always willing to sell to you or buy from you, he has a friend in Monty Hall and is willing to wait for you to talk with him.

In his turn, Monty is willing to give you information about your choices. Is this company stuffing inventory channels? Is that company being investigated by the SEC? Are those companies being run competently? Are the companies improving net margins year over year? It is as if Monty is giving you a peek behind all the doors.

All of this information, and more, is available. The Foolish investor is willing to take the time to gather some of it together before talking to Mr. Market. Need I say it? The more information gathered, the better and more informed will be your decisions.

Granted, despite all the information you gather, there will be times when what at first appeared to be a car turns out to be a goat and you lose. That is part of the risk of investing in stocks. However, by gathering information, you are certainly improving your odds of choosing correctly.

Lottery advertisements often say, "You can't win if you don't play." That is certainly true. But they fail to tell you that the risk of not winning the jackpot -- who really buys a ticket in the hopes of winning two dollars? -- is tremendous. No amount of information gathering will improve your odds.

For the stock market, too, you can't win if you don't "play." But pulling together information will definitely improve your odds of distinguishing the goats from the cars.