Let’s Make a Deal

Make your choice, adventurous stranger,
Choose your door, and bide the danger.
Or wonder 'til it drives you mad
What would have followed if you had.

-C.S. Lewis, from The Magician's Nephew

A friendly wave to all you Māori out there, and welcome. Today is International Candy Coated Day. One of the things we are instructed to do there is to list our ten favorite candies. I don't have a list prepared right now, but as I recall I did spend some time two years ago listing the 10 Best Candy Bars ever.

(And while I'm thinking about it, after eating Jelly Bellies for the first time I compiled my favorite 10 flavors of those little guys. And if you need to be offended (and I think that you do), we came up with a few more "I like my Jelly Bellies like I like my women…." jokes. Funny but kind of crude, so you're warned.

And over on Monkey Barn there's a preview of the new controversial movie September Dawn, as well as more gifts if you're looking for that certain something to get Hyperion for Hyperion Day. (And you should be, but even if no, go see anyway because the ideas are awesome and you can use them for someone less important than I.)


(posted with permission from 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.

Knowing which packets were where, I consumed a peach and suddenly a thought occured to me. What I could split myself in two and ask the Hyperion of a few minutes ago what the odds now were that the pineapple was on the bed (after one had been eaten).


Before you answer, let us recap: initially there were three packages: two peach and one pineapple. One fell on the floor, unknowable to me. THEN, one package on the bed was consumed, peach. What I am asking is the probability that the remaining package of dried fruit ON THE BED was PINEAPPLE at that time.


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?

August 21, 2007


Koz said...

Whats the probability that my brain is going to explode after reading this?

Han Solo said...

Never tell me the odds

Bear said...

(Warning: I put some tables in here and they might not be formatted correctly unless you widen the window enough to fit them.)

I understand the Monty Hall problem. You SHOULD switch doors.

However, you claimed that the Monty Hall problem is the same situation as the crispy fruit. I do not think this is true, and here is why: You stated that in the Monty Hall problem "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." That is correct. In the crispy fruit situation, though, you did NOT know that the packet that you chose to eat (i.e. the revealed door) was not the pineapple one (the grand prize door). Thus, in the crispy fruit scenario, you DID reveal it randomly. The packet that you revealed COULD have been the pineapple one. Contrast this to the Monty Hall problem, where the host is essentially revealing to you a packet that he knows is not the pineapple packet.

Saying that the probability of the pineapple being on the bed is 2/3 is saying that there are 2 out of 3 scenarios--or, equivalently, 4 out of 6 scenarios--where the pineapple is the remaining pineapple on the bed. Let's look at the possible sequences of events. There are 6 possible sequences:

Sequence Fruit Falls You Eat Fruit Remaining on Bed
(Contestant Picks) (Door Host Reveals) (Third Door)

1 Peach A Peach B Pineapple
2 Peach A Pineapple Peach B
3 Peach B Peach A Pineapple
4 Peach B Pineapple Peach A
5 Pineapple Peach A Peach B
6 Pineapple Peach B Peach A

Because you ate a peach and not a pineapple, sequences 2 and 4 are eliminated. We know that they did not happen. Thus, the remaining possible sequences are:

Sequence Fruit Falls You Eat Fruit Remaining on Bed
(Contestant Picks) (Door Host Reveals) (Third Door)

1 Peach A Peach B Pineapple
3 Peach B Peach A Pineapple
5 Pineapple Peach A Peach B
6 Pineapple Peach B Peach A

There are four sequences remaining. Only two of these sequences have pineapple remaining on the bed. Thus, the chances that the pineapple one is on the bed is 1/2.

In contrast, the Monty Hall problem relies on the fact that in sequences 2 and 4 you would have known that the packet you were about to eat was pineapple and then eaten the other packet on the bed, which was peach. In that situation you use prior knowledge to alter the sequences so they end up like:

Sequence Fruit Falls You Eat Fruit Remaining on Bed
(Contestant Picks) (Door Host Reveals) (Third Door)

1 Peach A Peach B Pineapple
2 Peach A Peach B Pineapple
3 Peach B Peach A Pineapple
4 Peach B Peach A Pineapple
5 Pineapple Peach A Peach B
6 Pineapple Peach B Peach A

In this situation the chances of the pineapple being on the bed after you've eaten your package are 2/3. However, you did not have this prior knowledge. In the Monty Hall problem, the host does.

Bear said...

Bah! Stupid formatting. Oh well. The people who care will copy and format the tables correctly.

Francis said...

I'm not sure how you ended up posting this today, but I was trying to explain this to LJS two nights ago.


Very random.

Dragon said...

So was the package on the floor, pineapple?

Bear said...

OK This is an attempt to clean up those tables. Just because I know Hype, at least, will look at them.

All 6 possible sequences:

Sequence__Fruit Falls____________You Eat________________Fruit Remaining on Bed
__________(Contestant Picks)_____(Door Host Reveals)____(Third Door)

1_________Peach A________________Peach B________________Pineapple
2_________Peach A________________Pineapple______________Peach B
3_________Peach B________________Peach A________________Pineapple
4_________Peach B________________Pineapple______________Peach A
5_________Pineapple______________Peach A________________Peach B
6_________Pineapple______________Peach B________________Peach A

Possible sequences after removing the 2 that we know did not happen (the ones where you eat a pineapple):

1_________Peach A________________Peach B________________Pineapple
3_________Peach B________________Peach A________________Pineapple
5_________Pineapple______________Peach A________________Peach B
6_________Pineapple______________Peach B________________Peach A

Hyperion said...

Koz - 7/5

Han Solo - At least your wife isn't into incest. Oh wait....

Bear - It is the same problem. I just was casual with my language. The essential facts are that a person is in a closed loop and that knowledge has taken out one of the options. Yes, I picked up one of the peach packets randomly, but as soon as I had knowledge that it was peach I eliminated it from my hypothetical situation and wondered what the odds then were. In essence I was Monty Hall and the contestant, but it does not change the actual probability for me.

Contrastingly, if my sister came in my room right then and I said "I have one package of Crispy Fruit on the floor and one on the bed with me. One is pineapple and one is peach." She would say there are 50-50 odds of the pineapple being on the bed, and she would be correct, if she were choosing. But since my initial "choice" was to think of the pineapple on the floor, the probability is still 2/3.

Francis - It's like the Universe is aligned to help you!

Dragon - That would just really confuse people!

Bear - I appreciate you putting up the tables, but I think the fact that I was trying to simplify the analogy for laymen got you caught up. Read my comment above; it is 2/3.

tiff said...

Still makes.No.Sense.

At all.

RC said...

(scratching head looking at tables)

Hyperion said...

I realize now I am going to have to do a follow up tomorrow. I also realize I was a bit "loose" with some of my descriptions in the name of readability, which doesn't help for people who are trying to literally follow. So show up tomorrow and all will make sense.

Or you will light yourself on fire.

Lady Jane Scarlett said...

It doesn't matter to me what kind of fruit dried stuff is on the floor, just as long as I get a nice view of those hot buns as you pick 'em up.
100% satissssfied...no statistics. Not exactly my best MENSA moment, maybe they'll kick me out. But your reasoning is sound Hypey.