A second impediment to good research is the natural tendency to over-estimate one's skill in assessing probabilities. One might think evolution would have selected by now for ability to maximize chances to get what one wants. To the contrary, we lack very advanced capacities for making choices critical to success in achieving our goals. The following example makes the point.

Suppose you are contestant on the television game show, "Let's Make a Deal." The emcee, Monty Hall, gives you the choice of one of three doors, explaining that behind one door is a grand prize. Your chances of winning the prize are 1 in 3. Suppose you choose Door A. Rather than opening it, however, Monty--who knows where the prize is--proceeds to open Door C. No prize. The prize is either behind door A or B. Monty now gives you a further option, to switch from door A to B. Should you do it?

Nearly everyone has the same response: It does not matter. The odds of winning are the same whether I stay with door A or change. We think the chances of winning with door A were 1/3 when we began, and are 1/3 now. And the chances of winning with door B were 1/3 then as well now, too. We might further reason that the chance of A being the winning door is 50%; and, likewise, the chance that B is the winning door is 50%.

Wrong. While most people find it virtually impossible to believe this is true, the fact is that these calculations are dead wrong. You will actually double your chances of winning by abandoning door A and changing to B. Under the circumstances, there is only a 33% chance the prize is behind A, but a 66% chance the prize is behind B.

Keith Devlin explains the puzzle as follows. "By opening his door, Monty is saying to the contestant "There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3."

As we seek systematic answers to our research questions, we must be on guard against over-confidence in our innate ability to assess probabilities.

PS. For those still not convinced, see Keith Devlin, "Monty Hall Revisited," accessed by Gary Comstock, 5/22/2007 at Devlin's Angle, December 2005.