The Hyperion Chronicles

#462 Let's Make a Deal

The other day I wrote a post over on Monkey Barn detailing my newfound appreciation for Crispy Fruit, a freeze-dried product that is the bomb-diggety (which is to say, good enough to make me use phrases worn out since Alf was still a Top 20 hit.

In that post I told how I had three packages of Crispy Fruit, two peach and one pineapple, and all with identical packaging except for the small picture on the front. I threw the three packages on the bed and then jumped on after, my prodigious frame causing one poor man overboard.

At that point, I idly wondered what the probability was of the package on the floor being pineapple. If you have followed me this far you probably guessed the probability was 1 in 3, or 1/3 chance the one on the floor was pineapple.

And correct you would be.

But here is where it gets interesting.

2/3.

As I expected, this was not a popularly accepted answer. Some were brave enough to opine in Comments, while others did not want to face public embarrassment but wrote me privately, basically saying, "I don't understand how the answer isn't 50-50 or ½."

I will say this: probability is often one of the most counter-intuitive fields. It just doesn't look or feel right. I assure you, however, that the answer is indeed that the probability of the pineapple being on the bed is 2/3. How can this be? For that understanding, perhaps we turn to a much more famous example: Monty Hall's Let's Make a Deal.

If you're not familiar with the show, the contestant is shown three doors, behind one of which is a fabulous prize, while the other two contain gag gifts or nothing at all.

Here, I found a picture:

Okay, so behind one of those doors is the following: 20 billion dollars, the ability to nail three celebrities without your significant other getting mad, and the ability to kill three celebrities (hopefully a different three) without ramifications. What more could you ask for?

Behind the other two doors, you have to spend eternity in a windowless room with Gilbert Gottfried, Nancy Grace and whatever celebrities you would have killed if you had won.

So you really want to win. I make the prizes this one-sided because I'm hoping to convince you to pick wisely and not crazily. Of course, if you just have a "thing" for the number 3 (ever since Sesame Street), there is not much I can do. But we are assuming you do not want to spend eternity listening to Gilbert Gottfried complain about Jafar while Nancy Grace interviews Al Franken; you want to pick smartly.

Let us further stipulate that A) the prize has been randomly placed and B) you do not know which door holds what. At this point, I think we can all agree that your chances, your probability of selecting correctly is one in three, or 1/3.

You choose Door #3. However, before I open that door, I make it interesting. Because I know which door leads to owning the Trail Blazers and trysts with Carre Otis, I open Door #2 to reveal Rosie O'Donnell and Rachel Marsden chatting amiably. I have purposely shown you a door that is incorrect.

Now you know it is not Door #2. Before I open your selection (Door #3), I offer you the choice to switch to Door #1. Do you take it? (If you are clever, you will notice that this is the same scenario as the Crispy Fruit situation, only my example is more fun to pay attention to because there's a good chance Gary Coleman will show up. (But in which door?))

Okay, as you probably feared, the answer is the same: you WOULD switch doors and take Door #1, at least if you understood probability. I feel several of you gnashing your teeth. Slutting up my analogy doesn't make it any easier to comprehend, but hopefully it affords me the opportunity to make things clear.

Back when you chose Door #3, what if I offered you the chance to take both
Door #1 and Door #2. I would open both doors, and if EITHER door had the cash and Keisha Castle-Hughes, you would win. Would you take it then?

Of course you would. You would be getting two doors instead of one, and no matter what bizarre attachment you have to the number 3, you can add.

Well, essentially that is what I am offering. I am still offering you Door #2 and Door #1. All I have done is reveal that Door #2 does not have the prize. I did this with knowledge. I did not open it randomly. I know it does not, which makes all the difference. The probability of the billions and the babes being in the remaining Door #1 is 2/3, because the original probability still holds.

Just to further baffle you: if I brought a new contestant up to play, and they have the two doors left to choose from, Door #1 or Door #3, the probability is 50-50 they will choose the right door. (Now your head really hurts, huh?)

I know it's tough, but because you are the same person choosing the door (or in the original example, the fruit), and because we are dealing with your choices and the knowledgeable removing of Door #2 from the options, the situation remains a closed-set of probability and you SHOULD switch doors.

Now, what is the probability everyone understood that?

Hyperion
August 21, 2007