There has been a recurring contention on a correct prediction by the genius columnist Maralyn Vos Savant in the parade magazine (a weekly insert in this newspaper). Back in 1990, she made a prediction about the odds for a particular game on the popular TV game show The Price is Right. The game show would give a contestant the choice to pick from 3 prizes hidden individually behind 3 curtains. One curtain would have a fantastic prize such as a brand new car and the other two might have a goat. After the contestant would pick one of the curtains, the rules of the game were to open one of the other curtains having a goat and then ask the contestant if they wanted to change their choice of curtains. No matter which choice you make, at least one of the other curtains has one of the undesirable consolation prizes so you are always left with two curtains, one having the new car and the other with the second consolation prize.
To recap, you have selected one out of three curtains and one of the curtains you did not select was opened to see it is an undesirable choice. You now can keep your original choice or switch to the remaining closed curtain in hopes that the correct choice is that curtain. This comes to the probabilistic dilemma, is it better to switch your choice or stick with your original choice. Apparently the popular decision is to say the probability is 50/50, either your curtain or the other curtain has the new car so it is with equal probability that either curtain has the car so it does not matter if you switch. You can choose to switch or you can keep your original choice and the odds remain the same. This however is the wrong interpretation and the correct answer given by Mrs. Savant, the odds are 2/3 in your favor (not 50/50) that it is always better to switch your choice. Again you may ask, “how can that be when you have only two choices: keep the curtain you have or switch, and only one of those two curtains has the grand prize” or, “how can the odds always be better by changing than to keep your current choice?” So although it is true that you are now left with two curtains, one being your original choice but the other being the only one left from those you didn’t choose, it really is more likely that the remaining closed curtain from those you didn’t select has the higher probability of containing the grand prize.
Consider an extreme example, let’s say there were originally a million curtains to choose from. When you make your choice, you have a million to one chance of selecting the correct curtain. This probability for your first choice is now set, the likelihood of that original selection is fixed as being very unlikely to have the correct choice. Now if all but one of the remaining 999,999 curtains are then opened, we now have an extreme example of the bias in the Price is Right game. You are still left with two choices for the grand prize, only one or the other has the prize and although here it should be clear that opening all the other curtains has made the remaining curtain almost certainly the correct choice. The same argument could be made with a hundred curtains or even ten curtains. By lowering the number of curtains involved in the game this way, the probability of obtaining the correct choice by switching does lower so that by the time you are down to only three curtains, the probability is only 2/3 that switching will give you the new car. Marilyn correctly described the math and subsequently much correspondence in opposition to her stance came in from the general public. Indeed, when I first heard the “riddle” I had disagreed with her but subsequently have been able to see the correctness in her position but have found the mental deliberations delightful. Hopefully you can get some cerebral exercise from this math question also.