This is the second part of a puzzle and paradox column that ran last Friday, and it’s taken from my tutorial on decisions.

The first part of this two-part series built a foundation by example for how to analyze the confusing Monty Hall paradox. The section will complete the analysis. Remember that these puzzles are more than just mathematical recreations. They go to the heart of how to make rational decisions based on data about a situation. Just as the probabilities can change radically in the Monty Hall paradox, the chances of having cancer based on field tests, or being a drug user based on urine samples can change radically depending on whether the various parameters become linked in some way.

With that preamble, here’s Monty… A guest on his television show gets to pick a door to open. There are three doors. Behind two of them is something of not much value. The third door hides a wonderful prize. The odds of picking the right door are obviously 1 in 3. However, to build more interest and maybe use up some extra time before a commercial, Monty does not immediately open the chosen door. Instead, he offers the contestant an interesting choice. Monty opens one of the unchosen doors and shows that it is not the prize door. That leaves only two closed doors. Now the zinger: Monty gives the contestant an opportunity to keep the chosen door or change to the unopened door. What should the contestant do to maximize the probability of winning? Stick or change?

Again, think about this before reading on and be prepared to defend your answer.

Well, if you are like most people, you will say that it makes no difference what the contestant does. When the game started, the odds were 1 in 3 that the winning door would be chosen, and through Monty’s generosity, those odds are now even. It doesn’t matter which door is opened, so there is no need to change. The contestant can keep the same door or change. Either way the odds of winning are the same. If you thought like that, you are wrong. The odds are 2:1 favoring change. You could win a lot of money at that.

I have a difficult time trying to understand just why some people get so incensed when I tell them the answer. Some people get rather emotional. So what went wrong? In essence, the common error is to look at the wrong thing. This is another example of human psychology getting in the way of solving problems which are simple for a machine.

Instead of thinking about the odds of winning, think about the odds of losing. The contestant starts with a 2/3 chance of losing. But of the two losers, Monty eliminates one. He did this by looking to see if the door he opened had the prize. By the act of looking, he linked the events just as in the case of guessing the gender of the children (see previous column or get the whole tutorial). With a little bit of thought, the odds of the other unopened door winning is the same 2/3. Therefore definitely change.

However, that explanation doesn’t seem to do much good for the confirmed sceptic. So consider a different approach. Label the doors W, L1, and L2 for winner and loser #1 or #2. Then the possibilities are simple.

If you chose W, then you lose by changing.
If you chose L1, then you win by changing.
If you chose L2, then you win by changing.

Since you don’t know which one you picked, of the three possibilities, you win twice by changing and once by holding. So change.

Okay, experience tells me that some people are still not convinced. The only answer for them is to play the game thousands of times and tabulate the results. This can be done with a simple program, but several groups have posted the game on the Internet. Here’s one that I happen to like (it uses a Java applet).

Log onto this site, enter the number of games you want to play (typically more than 1000 for good statistics), and it will give you the results if you always switch or if you never switch. Guess what? You should always switch.

Finally, there is the analysis I first used, so it might not mean much to you, but this is how a physicist might approach the problem. Instead of trying to solve the problem, change it to one with an obvious answer. Why limit it to three doors? Why not assume Monty had a million doors? Play the game as before, only now Monty opens all the unchosen doors except for one. Should you stay with your 1 in a million door, or change to the 1 in 2 door? The answer is obvious.

Sometimes when faced with what seems to be a complex problem, one can simplify it in this way by considering extreme values.

A classic example of this approach is the old puzzle about drilling a hole through a sphere to make a bead. This puzzle can easily be solved with the available information. Assume the hole is 1 inch long, what is the volume of material left in the sphere after the hole is drilled? The answer will be given later – maybe.