Brain Teaser/Logic Problem (2 Viewers)

[gopherblue]

I think this brain teaser/logic problem is well-suited to this crowd, in that we are usually thinking about probabilities, EV and game theory...

You're on The Price Is Right, and for your game, you're given the choice of three doors:

Behind one door is Trihonda's card protector; behind the two others, bupkes. You pick a door, let's say Door 1, and Bob Barker, who knows what's behind the doors, opens one of the other doors, let's say Door 3, which has bupkes. He then says to you, "Do you want to change your pick to Door 2?"

Do you switch?​

Note: While we live in a google world, don't hit the interwebz for the answer to this until after you've chosen and posted a response with your rationale. I immediately knew the answer as a gabooler, but I work with a lot of smart folks who spend a lot of time crunching numbers, and I can tell you that there is significant disagreement as to what they think is the right answer.

 

[jbutler]

I would hope that anyone who gambles seriously would already understand the solution to this question, but I do remember when I first confronted it years ago, it took significant explaining (this was pre google).

 

[Forest@Farmington]

I know the answer and it is from one of my favorite movies.

 

[bentax1978]

I loves these. This particular problem is better know by the name of the host of another game show from back in the day, Monte Hall. I remember this one well from my game theory classes back in college so I won't spoil anything by posting until other have answered.

 

[H|Q]

Do you know where the chip is?

 

[gpc]

Originally I had a 33% chance and now I have a 50% chance but the contents did not change. Making the switch would mean I chose correctly twice in a row so my odds are actually worse. I will stay pat.

 

[JFCJ]

Do you know where the chip is?

I know where the chips are...

 

[gopherblue]

Do you know where the chip is?

Not in this scenario. But Bob Barker does.

 

[jbutler]

Originally I had a 33% chance and now I have a 50% chance but the contents did not change. Making the switch would mean I chose correctly twice in a row so my odds are actually worse. I will stay pat.

Remember that the chip was placed behind a certain door from the beginning and can't be moved after you've selected the first door.

With that in mind, consider whether your odds are actually 50/50 at this point. If you think they are, then what happened to the likelihood that the chip was behind the door that Bob opened to reveal nothing? Was there never any chance that the chip was behind the door that was already opened?

 

[Ronoh]

Originally I had a 33% chance and now I have a 50% chance but the contents did not change. Making the switch would mean I chose correctly twice in a row so my odds are actually worse. I will stay pat.

If you stand pat you have your initial 33% chance of winning. If you switch you have 66% chance of winning.

If you pick door #1 you win if the prize is behind door #1. If you switch you will win if the prize is behind door #2 OR door #3. It's that simple.

 

[acl1904]

The way my luck has been running as of late, I switch and I win bupkes...

 

[gpc]

I googled and am more confused but glad I know I am not the only one and I learned the definition of bupkes.

 

[Ronoh]

I googled and am more confused

No confusion necessary. The host always eliminates one of the doors you did not initially choose. If you choose door #1 with the intention of switching there are only three possible outcomes.

The prize is behind door #1:
The host will show you bupkes behind door 2 or 3 (doesn't matter which one) and you get the bupkes behind the other one - LOSER.

The prize is behind door #2:
The host will show you bupkes behind door 3 and you switch to door 2 - WINNER.

The prize is behind door #3:
The host will show you bupkes behind door 2 and you switch to door 3 - WINNER.

There are no other possible scenarios.

 

[bivey]

make the switch. always bet on the nutz.

 

[spikeithard]

'21' answered this one for me.

 

[gopherblue]

Remember, there is no spoon.

 

[p5woody]

If you stand pat you have your initial 33% chance of winning. If you switch you have 66% chance of winning.

If you pick door #1 you win if the prize is behind door #1. If you switch you will win if the prize is behind door #2 OR door #3. It's that simple.

Yep, I agree with this.

 

[pltrgyst]

^ Yeah. Old stuff. You always switch.

 

[Anthony Martino]

Seen this one a long time ago, switch is the way to play

 

[phaze12]

i read this one a while ago. like bentax said it was named the monte hall something or other. if you have read poker books you should already know the answer and why.

 

[jbutler]

I definitely agree with @Ronoh's statement, but that wasn't enough for me to understand it when I was first presented with this question way back when. The way it was first explained to me was by what I take to be the theoretical basis and it took me a while to "get" it. Maybe I'm slow, but it took a different way for me to understand it more fully.

The segment of the below video that begins at about 2:45 is essentially the same way a friend eventually made me understand.

 

[gpc]

The ace of spades out of a deck of cards explanation makes sense but I'll admit the three doors has me stumped.

What if this was a casino game and the prize was $100. The dealer will reveal one of the bupkes. What is the most you will wager?

 

[jbutler]

The ace of spades out of a deck of cards explanation makes sense but I'll admit the three doors has me stumped.

What if this was a casino game and the prize was $100. The dealer will reveal one of the bupkes. What is the most you will wager?

Assuming you must wager in even dollar amounts, $66.

 

[MoscowRadio]

I guess the thing I don't understand about this problem is this: if Bob opens door #3 then there's now a 2 in 3 chance the prize is behind door #2. However, couldn't the odds be reversed and now door #1 now holds a 2 in 3 chance that the prize is behind your original choice of door #1?

This is actually the first time I've encountered this problem, so I apologize if I sound totally naive.

 

[manamongkids]

I guess the thing I don't understand about this problem is this: if Bob opens door #3 then there's now a 2 in 3 chance the prize is behind door #2. However, couldn't the odds be reversed and now door #1 now holds a 2 in 3 chance that the prize is behind your original choice of door #1?

This is actually the first time I've encountered this problem, so I apologize if I sound totally naive.

http://math.ucsd.edu/~crypto/Monty/monty.html

 

[jbutler]

I guess the thing I don't understand about this problem is this: if Bob opens door #3 then there's now a 2 in 3 chance the prize is behind door #2. However, couldn't the odds be reversed and now door #1 now holds a 2 in 3 chance that the prize is behind your original choice of door #1?

This is actually the first time I've encountered this problem, so I apologize if I sound totally naive.

http://math.ucsd.edu/~crypto/Monty/monty.html

The game is fun, but the explanation you can click through to get to from that page is pretty good.

Here's the link directly for convenience and in case it gets overlooked once people go through just to play the game.

 

[12thMan]

The explanation that really hit home for me was this: Pretend he didn't open the door to show you nothing, but instead after you made your selection he said "Okay, you have chosen door #1, I'll let you keep that selection or you could have both of the other doors.", most people at that point will say "Obviously I would take the other two" and it's then easily explained that by Monty showing you whats behind one door and offering the other it's exactly the same as you being offered the two up front.

 

[jbutler]

The explanation that really hit home for me was this: Pretend he didn't open the door to show you nothing, but instead after you made your selection he said "Okay, you have chosen door #1, I'll let you keep that selection or you could have both of the other doors.", most people at that point will say "Obviously I would take the other two" and it's then easily explained that by Monty showing you whats behind one door and offering the other it's exactly the same as you being offered the two up front.

Best explanation I've heard yet.