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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  Comments (6) un-categorized
  1. Comment by: Zack Estes   (3/1/10 12:44 PM)   

    First things first.  Jimmy Buffet wrote the song referenced in the title (I may need those five points later).

    And now time to tackle the topic at hand.  Yes, the contestant should switch doors and should always switch doors because by switching doors you are doubling your chances of winning.  Lets break down the Monty Hall Paradox and see why you should switch.

    First off, for this explanation to be true we have to make the assumption that the host knows what is behind each door and that he will always open a door with a Zonk after the contestants makes their initial pick and gives them the option to switch doors.

    If the contestant decides not to switch doors after their initial pick they have a 33% chance of selecting the riches of Xanadu.  What Monty does after that doesn't really matter because the contestant already made up their mind about switching.

    Now lets look at it a different way.  The chances of selecting a Zonk on your initial pick is twice that of selecting the riches of Xanadu.  Since Monty must open a door containing a Zonk after you initial selection your chances of switching doors and winning all the riches of Xanadu are twice that of not switching.

  1. Comment by: Mike Jacka   (3/1/10 01:22 PM)   

    Good answer Zack.  Unfortunately, I just posted my next update, so you won't get any credit in my next posting.  But you got this one right.  (By the way, as you'll also see from my update, you only get 2 and a half points. But we can always use every point someone is willing to give us, can't we.)

  1. Comment by: Ben   (3/3/10 02:09 PM)   

    I disagree . . . Monty is ALWAYS going to show a zonk before the final results.  So, given that consideration, the chance of winning the "Riches" appears to be 50-50 in that the contestant either picked the winner or a zonk . . . the Monty-revealed "zonk" (customary in this game) is a "freebie" .  HOWEVER, we all know this 50-50 chance is not true -- the contestant is picking a 1 of 3 winner so the initial odds are 1 / 3.  In reality, a "zonk" being shown means nothing.  Residually, there will always be that third door that was neither selected nor contains the BIG PRIZE.  Switching your selection does NOT increase your chances of winning -- they are still the initial 1/3.  The elimination of the third door gives the illusion of increasing your chances, but the initial selection was 1 of 3 which has not changed.  Giving a second chance to trade doors provides the contestant a sort of "modified" chance now at 1 of 2.  BUT, s/he could already have selected the winner. 

    Let's assume the contestant initially selects the grand prize unbeknownst to him/her.  Monty shows the third-door "zonk" and now two doors remain.  In this case, if the contestant switches s/he loses the grand prize.  Conversely, should the contestant have initially selected one of the two "zonks", Monty reveals the "third-door "zonk" and again, there are two doors remaining.  If the contestant decides to switch, s/he goes from zonk to winner of the grand prize. 

    Bottom line:  The odds have seemingly improved to 1 in 2, but if in reality that is a betterment of from 33.3% to 50%, it still does not represent " . . . your chances of switching doors and winning all the riches of Xanadu are twice that of not switching." . . .

  1. Comment by: William Johnson   (3/5/10 03:42 PM)   

    Your chances of winning are the same, no matter what you do. However the value of your original choice has of course went up. It's the Deal or No Deal Correlary of the paradox. (I'm making that one up.) If you assume that at the beginning of the game your choice has a value of RoX (and I'd like to go with the Xanadu of Charles Kane myself) + Zero, (since the Zonk is probably zero, unless you're a masochist, and since this is probably an internal auditor as the contestant, then masochism is most probably a given.) divided by 3, the value can be represented as RoX/3. Now the value is RoX/2. So your choice is worth more now, but your chances of winning do not change at all.

    You can't observe it. Therefore your chance haven't improved. However the cat inside the one zonk may be dead since Mr. Schrodinger was recently put in charge of game design for the new improved let's make a deal.

  1. Comment by: Mike Jacka   (3/7/10 09:26 PM)   

    Bill (and I can call you Bill because I know who you are, and I can say the following because I know who you are), you definitely need help.

  1. Comment by: William Johnson   (3/10/10 10:25 AM)   

    Sad part is that I was wrong... I played fair and didn't read the answer. Oh well. At least my answer has the value of having never been tried. Ah well if I could have only reprogrammed the simulator.

    Yes I do need help!

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