Some puzzles find their charm in unexpectedly coupled events. Events can become coupled or intertwined such as to radically change the probability of a result in unexpected ways. To this day, learned people argue over the Monty Hall paradox. Most people get it wrong.
Remember when we discuss paradoxes like the Monty Hall, we are essentially considering aspects of psychology. If we program a computer to make the calculations correctly, it will not be bothered by any paradox. The key word here is “correctly.” Some of these puzzles and paradoxes are so subtle that they can bias how a person programs a solution.
First, we start with a puzzle that is related to the Monty Hall paradox, but is much easier to understand. Assume that, on the average, are equal numbers of boys and girls are born in the world. I have two children. At least one of them is a boy. What is the probability that the other is a boy?
Before reading ahead, be sure of your answer and be prepared to defend it. You might want to re-read to conditions.
By the conditions given above, my children could have been in order of age from younger to older: BB, BG, GB, GG. However, I said at least one was a boy, so GG is ruled out. Of the remaining choices, two have the second child being a girl and only one has the second child being a boy. Therefore the odds are 2:1 the other child is a girl. The wording of the puzzle linked the two children such that their genders were no longer independent events.
Okay, now let’s change the problem slightly. Suppose I have two children and the older one is boy. That condition immediately rules out two of the possibilities listed above, and in the two that remain, the odds are even that the younger child is a boy or girl. By identifying one child specifically, the genders are kept independent and the odds remain even.
Try the same thing with two types of coins, say a quarter and a nickel. Flip them and report that at least one is a heads. What is the probability the other is heads? Many people will naively say the odds are even. You can make a lot of money on this, and in fact, some carnival scams are based on exactly this type of paradox of linking probabilities.
Monty Hall: If you fully understand that example, you are ready for 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, 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. One of these two hides the wonderful prize. The other is a nothing. Now comes 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 and be prepared to defend your answer. You are probably wrong. The correct analysis will be given in another column. Since many readers will not believe the logical analysis, I will also include a computer-based solution that you can run yourself.
This column was taken from my tutorial on decisions, which is available here.