Makes Sense to Me (My Whole World Lies Waiting Behind Door Number 3)
First things first. Five points to the first to identify who wrote the song referenced in the title.
Next things next. Ben caught me. In my previous post I fell into my own mental tunnel and didn’t carry the decimal point. I wrote that 70 percent of 40 was 2.8 rather than 28. It doesn’t change the conclusion. However, I should be able to do simple math.
Anyway, on to our three prisoners dilemma. To recap — three prisoners are to be executed. The tyrant determines one will be spared, but only tells the guard. Prisoner C asks the guard which of the other two will be executed. The guard tells him Prisoner A. Prisoner C now believes his odds of being spared have moved from 33 percent to 50 percent. Is he correct?
Well, if you’ve been following along, you probably have guessed that I’m going to say that the prisoner is incorrect. However, there are also a few of you who think I’m wrong. Let me see if I can convince you. (And, again, I am relying heavily on Massimo Piattelli-Palmarini’s book Inevitable Illusions for this explanation.)
The correct answer is that Prisoner C’s chance of being spared has not changed, but the information provided by the guard increases Prisoner B’s chances to 66 percent. Here’s why. Before we are given the additional information, we know that the cumulative probability of Prisoner A and Prisoner B being the one that is spared is 66 percent. Keep in mind, a cumulative probability does not determine the individual probabilities; rather, the probability of either Prisoner A or Prisoner B being spared is 66 percent. As soon as we are given the information that Prisoner A will be executed, the cumulative probability (66%) passes to Prisoner B. Think of it this way. If the guard knows Prisoner C is being shot (66 percent of the time) he is forced to say A or B. The other 33 percent of the time, the answer A or B is given because Prisoner C is being spared. Sixty-six percent of the time the answer is based on him being shot, 33 percent of the time the answer is based on him not being shot. Sixty-six percent of the time, he will be shot, no matter what answer is given.
Three arguments are often raised: 1) the cumulative probability of all three prisoners is 100 percent, so removing one of them means the cumulative effect should be divided between the two remaining prisoners; 2) if the argument from the previous paragraph is correct, then it must mean that, upon the execution of prisoner A, prisoner B’s chances increase to 66 percent; and 3) because the cumulative probability for A and C is 66 percent, revealing that A will be shot transfers the cumulative probability of 66 percent to C. You can use the comments section below to refute these arguments (or support them if you believe they are true).
And now, let’s move on to the granddaddy of all mental tunnels — The Monty Hall Paradox. This is based on the old game show Let’s Make a Deal. (And if you’re too young to know what I mean — well then just Google it and get off my lawn.)
Three prizes are available behind doors one, two, and three. One contains the riches of Xanadu (the one associated with Kubla Khan, not Olivia Newton-John). Two have Zonks (that is, horrible, worthless prizes). Our contestant (dressed as a funny internal auditor — think big glasses, green visor, giant pencil behind the ear) chooses door number one. Monty (our host for the festivities), then has Carol Merrill reveal what is behind door number two. It is a Zonk. He offers our contestant the opportunity to change from door number one to door number three. Should the contestant switch? Let the games begin. And don’t forget to show your work.
Posted on Feb 24, 2010 by Mike Jacka
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