The Two-Envelope Problem Solution: Part I

http://wmbriggs.com/blog/?p=2645 ^ | William M. Briggs

[mattstat]

Another probability “paradox”, the two-envelope problem1, goes like this:

Before you are two envelopes, A and B. One of them contains $X and the other $2X (which is equivalent to $Y/2 and $Y). You pick one envelope and are (1) asked if you would like to keep it or switch, or (2) open it, view its contents, and then asked if you would like to keep it or switch. Which strategy, keeping or switching, is likeliest to win you the big bucks?

No peeking solution

The traditional paradoxical solution to (1) is to argue this. Suppose you pick A, which can be said to have $X (recall you do not peek). There is then a 50% chance that B contains 2X (dropping the dollar signs, but understanding they are still there, lurking) and a 50% chance that B contains X/2. The “expected value“2 of B is said to be

0.5 * 2X + 0.5 * X/2 = 5/4 X.

So clearly you should switch, since the expected value of the envelope you did not pick is larger than X, which is the value of the envelope you hold.

But that’s nuts, because you could have picked up B, in which case the expected value of A is 5/4 X, too. Oh, what to do!...

[E. Pluribus Unum]

Same thing as the Monty Hall 3-door problem.

[shibumi]

Option 3:

Pass on the bait offered by the guy pimping wmbriggs.com and move on to the next legitimate article.

[bcsco]

I ate that pig at a roast on Saturday. Now there’s only two doors...

[tacticalogic]

If you get to look first, then if the first envelope contains an odd amount, you probably want to switch.

[Vermont Lt]

If there is no peeking, how will you know if you have chosen correctly.

Moral of the story: if someone offers you free money, just take it.

[Jack Hammer]

The solution, of course, is to ‘cut the Gordian knot’ - that is, you grab both envelopes, slug the guy playing the silly probability game, and walk away.

[bcsco]

Moral of the story: if someone offers you free money, just take it.

I did. I ate the pig on Saturday and it didn't cost me a cent.

[Sam's Army]

"Pass on the bait offered by the guy pimping wmbriggs.com and move on to the next legitimate article."

I love the smell of napalm in the morning.

[shibumi]

So do I.

Smells like ..... victory.

[Vermont Lt]

I am going to my first pig roast Wed. I am really looking forward to it.

[bcsco]

Good, enjoy yourself. The one I attended has been going on for about 31 years as I’m told. About 150 people there. And the pig was only the centerpiece of a bunch of great, home cooked, food including mucho salads and pastries.

[Zeppelin]

He’s right in that you SHOULD switch.

The expected value of 5/4X is not illogical.

Think of it like this. Let’s say you choose an envelope and it contain’s $100. According to the riddle, you have a 50% chance of it being $200 and a 50% chance of it being $50.

I would say to switch b/c if you get the $50, then you’ve only “lost” $50, whereas if you get the $200, you’ve “gained” $100. And if you have an equal chance of gaining $100 as you do of losing $50, then go for it!

If the issue was such that you had a 50/50 chance of doubling or losing it all, then the expected value would be 1X and it wouldnt matter which you chose (from a probability standpoint).

[frithguild]

Sounds like what we all do in an voting booth.

[Sloth]

But why wouldn’t you reason the same way about your new envelope after switching?

[Amagi]

Man in Black: All right. Where is the poison? The battle of wits has begun. It ends when you decide and we both drink, and find out who is right... and who is dead.
Vizzini: But it's so simple. All I have to do is divine from what I know of you: are you the sort of man who would put the poison into his own goblet or his enemy's? Now, a clever man would put the poison into his own goblet, because he would know that only a great fool would reach for what he was given. I am not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool, you would have counted on it, so I can clearly not choose the wine in front of me.
Man in Black: You've made your decision then?
Vizzini: Not remotely. Because iocane comes from Australia, as everyone knows, and Australia is entirely peopled with criminals, and criminals are used to having people not trust them, as you are not trusted by me, so I can clearly not choose the wine in front of you.
Man in Black: Truly, you have a dizzying intellect.
Vizzini: Wait til I get going! Now, where was I?
Man in Black: Australia.
Vizzini: Yes, Australia. And you must have suspected I would have known the powder's origin, so I can clearly not choose the wine in front of me.
Man in Black: You're just stalling now.
Vizzini: You'd like to think that, wouldn't you? You've beaten my giant, which means you're exceptionally strong, so you could've put the poison in your own goblet, trusting on your strength to save you, so I can clearly not choose the wine in front of you. But, you've also bested my Spaniard, which means you must have studied, and in studying you must have learned that man is mortal, so you would have put the poison as far from yourself as possible, so I can clearly not choose the wine in front of me.
Man in Black: You're trying to trick me into giving away something. It won't work.
Vizzini: IT HAS WORKED! YOU'VE GIVEN EVERYTHING AWAY! I KNOW WHERE THE POISON IS!
Man in Black: Then make your choice.
Vizzini: I will, and I choose - What in the world can that be?
Vizzini: [Vizzini gestures up and away from the table. Roberts looks. Vizzini swaps the goblets]
Man in Black: What? Where? I don't see anything.
Vizzini: Well, I- I could have sworn I saw something. No matter.First, let's drink. Me from my glass, and you from yours.
Man in Black, Vizzini: [Vizzini and the Man in Black drink ]
Man in Black: You guessed wrong.
Vizzini: You only think I guessed wrong! That's what's so funny! I switched glasses when your back was turned! Ha ha! You fool! You fell victim to one of the classic blunders - The most famous of which is "never get involved in a land war in Asia" - but only slightly less well-known is this: "Never go against a Sicilian when death is on the line"! Ha ha ha ha ha ha ha! Ha ha ha ha ha ha ha! Ha ha ha...
Vizzini: [Vizzini stops suddenly,his smile frozen on his face and falls to the right out of camera dead]
Buttercup: And to think, all that time it was your cup that was poisoned.
Man in Black: They were both poisoned. I spent the last few years building up an immunity to iocane powder.

[mattstat]

Gee, shibumi, sorry you feel that way. Have you discovered a flaw in the solution?

Is “pimping” a term used only for non-commercial sites?

[FroggyTheGremlim]

"If you wrestle with a pig in the mud, you both get dirty and then the pig wins."

(or something like that)

- Senator John McCain

[mattstat]

It’s similar, but not identical to Monty Hall.