The "Let's Make a Deal" Paradox

Pete · 05-25-2003, 08:55 AM

OK - This is driving me bananas so I'm challenging you guys to explain this to me so I can convince myself that it's really true.

I was listening to NPR yesterday (weekend edition saturday) and they interviewed some math guy who explained the probability of Let's Make a Deal. The contestant chooses one of three doors, then Monty reveals an empty door and gives the contestant a chance to change their choice. It seems obvious to me that the chances at this point are 50/50. Even chances for switching or not switching. This guy says that's not the case. The chance of getting the right door if you switch is double that if you don't switch (2/3 to 1/3). But he didn't go on to explain any further. ARRRGH

So this morning I woke up Griff cause it was driving me crazy. Then I went online to look it up and found this link. I think I understand the way thry're explaining it but my intuition is still fighting me.

Basically, your chances of guessing the wrong one are 2/3. If you guess the wrong one, Monty must pick the other wrong one so by switching, you get the right one. I guess it all boils down to the fact that you decrease Monty's choices by picking the wrong one. How could you explain this to someone without p-ing them o?

Pete · 05-25-2003, 09:06 AM

Now let me answer my own question. I think I like this one. if we just increase the scale, it seems much more obvious. We have 100 lottery tickets (and one is the winner). I choose one. The probability that one of the remaining 99 is the winner is 99/100. Then someone removes all but one of those 99 saying that they are all losers. Then it seems obvious that you'd want to switch. I don't have the exact math at my fingertips for this one but my intuition feels much better about it. sigh.

BTW here's where I found this.

Undertoad · 05-25-2003, 10:51 AM

We did this one on the Cellar mk 2!

xoxoxoBruce · 05-25-2003, 09:35 PM

How do you know the applet doesn't cheat?

arz · 05-27-2003, 03:12 PM

Well, strangely enough in 5 runs of the applet I stayed with my original guess all five times and won the $ 4 of those 5...

5 runs is not statistically valid and yet the initial intuition answer is reinforced not disproved.

Update:
I ran the simulation 12 times; 6 times without switching and 6 times with switching. Result: I won 4 times out of 6 in either strategy.

Undertoad · 05-27-2003, 03:32 PM

Applet schmapplet. I'm willing to make you a deal right now.

You put down $100, and I'll put down $500.

I'll lay out 52 cards, face-down, and I'll look at them while I lay them down. You have to pick the 3 of clubs. You pick your first card, but don't turn it over. I'll take away 50 other cards and show them to prove that they aren't the 3 of clubs. Now if you have the 3 of clubs, you win the pot, while if I have the 3 of clubs, I win the pot.

Agreed?

xoxoxoBruce · 05-27-2003, 04:50 PM

Quote:

and I'll look at them while I lay them down.

Methinks this be the fly in the ointment. The odds are 51 to 1 against me. No thanks.
After you remove the 50 cards, let me choose which of the 2 I want. And one of them is the 3 of clubs, your on.

arz · 05-27-2003, 05:04 PM

The point of my previous comment is that the demonstration applet doesn't help reinforce the argument on the website since most people won't run it a statistically valid number of times to confirm the non-intuitive strategy.

Were I that guy I'd remove it since it's not helping.

Pete · 05-27-2003, 08:16 PM

Quote:

Originally posted by arz
... most people won't run it a statistically valid number of times to confirm the non-intuitive strategy.

I, for one (not a statistically valid sample) felt compelled to run the applet a statistically valid number of times. Maybe you should try Vegas cause I think you just got lucky.

Hey Toad - Thanks for the card example - that's a good one. I think my intuition is catching on. Griff's Dairy farmer uncle will like this one at the next family gathering. He asked me one Thanksgiving what was the maximim amount of money you could make playing Jeopardy. Cool guy. Can't remember the answer tho.

SteveDallas · 05-27-2003, 10:36 PM

Ummm... let's see...
You'd need to get all the questions in Jeopardy except a $100 one... that would be $8900. The last $100 question would be the Daily Double... you'd bet the whole shebang and enter Double Jeopardy with $17800.

You'd then proceed to get all the Double Jeopardy questions except for two $200 ones. That would be $17600, added to your take from the first round would give you $35400.

You'd then hit the penultimate question, a Daily Double, bet it all, and have $70800. And then the last question, the final Daily Double, another bet of everything, and you enter Final Jeopardy with $141600.

In Final Jeopardy, you bet it all once again and leave with a cool $283200.

But that's the best possible placement of the Daily Doubles. (I'm not even sure if they ever put them in the top row.) If you had the worst case scenario, you'd hit it on a $500 question and it would be your first question... so you'd end up with $500 after answering it correctly, then run the table, and enter Double Jeopardy with $9000. You'd then hit the two daily doubles under $1000 spaces as your first two Double Jeopardy questions, and have $36000. All the rest of the squares are worth $16000 for a total of $52000. Then you have $104000 after winning your Final Jeopardy bet.

What did I miss??

BrianR · 05-28-2003, 05:48 PM

TAXES!!!!!!

Brian

hot_pastrami · 06-13-2003, 11:45 AM

For me, the "Let's Make a Deal" question took a few minutes of thinking the first time I heard it before I grasped the way the odds work. None of the explanations above would have helped me much though, so I thought I'd offer an explanation from a different angle for those whose brains work like mine.

It helps to turn the question around in your head a little bit... let's assume there are two players, one who plans to stick with their original door, and one who plans to switch.

The one who sticks with their original door is effectively trying to pick the right door from the getgo, and has a one-in-three chance of being right. But the player who intends to switch doors is basically trying to pick one of the wrong doors from the getgo (since they don't intend to stick with that door), and this player has better odds... a two-out-of-three chance of successfully identifying a wrong door.

Even if the players don't decide whether they're switching until the question is posed, the odds don't change.... if you switch, then your original selection was effectively trying to identify one of the wrong doors, and your chances of successfully selecting a wrong door is about 33.3% better than a player who sticks with their one-in-three chance of guessing the right door originally.

Make sense? It can be a hard one to wrap the brain around.

vsp · 06-13-2003, 01:15 PM

And it has never made sense to me, for a simple reason:

I pick Door #1. Monty opens Door #2, and Rosie O'Donnell is standing behind it, so obviously I didn't want that one. He offers me the choice of keeping Door #1, or switching to Door #3.

What I do not, and probably will never understand is how the second choice is not _independent_ from the first one. Yes, Monty has information that I do not as to which door has the prize -- but the mere fact that I do not have that information makes it irrelevant to my choosing process. I don't know whether Monty is trying to help me or screw me over by making his offer, and I have no way of finding out short of making a blind choice; all I _do_ know is that between my first choice and my second, the situation has changed and my options are different.

When I make my second choice, I am making an independent choice between two doors. One has a prize. One has a goat. A third door has a goat, but that door is _no longer an option_, so I don't grasp why it is figured into the odds when I choose again. (Really, if Monty always offers you the choice to switch, the first choice is irrelevant; he will always open a goat door, so you'll always have a final choice that narrows it down to two.)

I've taken math courses through Calculus 3. I've taken Probability and Statistics courses. I've read lots of explanations of the "real" odds for each door... and I _just don't get it._

wolf · 06-13-2003, 01:21 PM

I don't get it either, but such a thing does not prey upon my mind. I, personally, might be excited to win the family of goats.

My enjoyment of the show was to see what goofy costumes the contestants were wearing, and I also loved the segment where Monty would look around the audience for an assortment of bizarre objects that women might have in their purses ... "I'll give you $50 for a hot dog ..."

russotto · 06-13-2003, 02:44 PM

In the two player version they ran sometimes, it was best to _NOT_ switch. The way that one worked, two contestants would each pick a door. Monty would then open a losing door which belonged to one of the contestants, and offer the other contestant the chance to switch to the remaining door.

05-25-2003, 09:35 PM	#4
xoxoxoBruce The future is unwritten Join Date: Oct 2002 Posts: 71,105	How do you know the applet doesn't cheat? __________________ The descent of man ~ Nixon, Friedman, Reagan, Trump.

05-27-2003, 05:04 PM	#8
arz Hand-of-Kindness Extender Join Date: Sep 2002 Location: So Cal Posts: 137	Point: Missed The point of my previous comment is that the demonstration applet doesn't help reinforce the argument on the website since most people won't run it a statistically valid number of times to confirm the non-intuitive strategy. Were I that guy I'd remove it since it's not helping.

05-27-2003, 10:36 PM	#10
SteveDallas Your Bartender Join Date: Jan 2002 Location: Philly Burbs, PA Posts: 7,651	Ummm... let's see... You'd need to get all the questions in Jeopardy except a $100 one... that would be $8900. The last $100 question would be the Daily Double... you'd bet the whole shebang and enter Double Jeopardy with $17800. You'd then proceed to get all the Double Jeopardy questions except for two $200 ones. That would be $17600, added to your take from the first round would give you $35400. You'd then hit the penultimate question, a Daily Double, bet it all, and have $70800. And then the last question, the final Daily Double, another bet of everything, and you enter Final Jeopardy with $141600. In Final Jeopardy, you bet it all once again and leave with a cool $283200. But that's the best possible placement of the Daily Doubles. (I'm not even sure if they ever put them in the top row.) If you had the worst case scenario, you'd hit it on a $500 question and it would be your first question... so you'd end up with $500 after answering it correctly, then run the table, and enter Double Jeopardy with $9000. You'd then hit the two daily doubles under $1000 spaces as your first two Double Jeopardy questions, and have $36000. All the rest of the squares are worth $16000 for a total of $52000. Then you have $104000 after winning your Final Jeopardy bet. What did I miss?? Last edited by SteveDallas; 05-27-2003 at 10:44 PM.

05-28-2003, 05:48 PM	#11
BrianR Cleverly disguised as a responsible adult Join Date: Jan 2001 Location: Dallas, TX Posts: 3,338	what you missed... TAXES!!!!!! Brian __________________ Never be afraid to tell the world who you are. -- Anonymous

06-13-2003, 11:45 AM	#12
hot_pastrami I am meaty Join Date: Dec 2001 Location: Salt Lake City, UT Posts: 1,119	For me, the "Let's Make a Deal" question took a few minutes of thinking the first time I heard it before I grasped the way the odds work. None of the explanations above would have helped me much though, so I thought I'd offer an explanation from a different angle for those whose brains work like mine. It helps to turn the question around in your head a little bit... let's assume there are two players, one who plans to stick with their original door, and one who plans to switch. The one who sticks with their original door is effectively trying to pick the right door from the getgo, and has a one-in-three chance of being right. But the player who intends to switch doors is basically trying to pick one of the wrong doors from the getgo (since they don't intend to stick with that door), and this player has better odds... a two-out-of-three chance of successfully identifying a wrong door. Even if the players don't decide whether they're switching until the question is posed, the odds don't change.... if you switch, then your original selection was effectively trying to identify one of the wrong doors, and your chances of successfully selecting a wrong door is about 33.3% better than a player who sticks with their one-in-three chance of guessing the right door originally. Make sense? It can be a hard one to wrap the brain around. __________________ Hot Pastrami!

05-25-2003, 08:55 AM	#1
Pete Resident President Join Date: Apr 2003 Posts: 81	The "Let's Make a Deal" Paradox OK - This is driving me bananas so I'm challenging you guys to explain this to me so I can convince myself that it's really true. I was listening to NPR yesterday (weekend edition saturday) and they interviewed some math guy who explained the probability of Let's Make a Deal. The contestant chooses one of three doors, then Monty reveals an empty door and gives the contestant a chance to change their choice. It seems obvious to me that the chances at this point are 50/50. Even chances for switching or not switching. This guy says that's not the case. The chance of getting the right door if you switch is double that if you don't switch (2/3 to 1/3). But he didn't go on to explain any further. ARRRGH So this morning I woke up Griff cause it was driving me crazy. Then I went online to look it up and found this link. I think I understand the way thry're explaining it but my intuition is still fighting me. Basically, your chances of guessing the wrong one are 2/3. If you guess the wrong one, Monty must pick the other wrong one so by switching, you get the right one. I guess it all boils down to the fact that you decrease Monty's choices by picking the wrong one. How could you explain this to someone without p-ing them o?

05-25-2003, 09:06 AM	#2
Pete Resident President Join Date: Apr 2003 Posts: 81	Now let me answer my own question. I think I like this one. if we just increase the scale, it seems much more obvious. We have 100 lottery tickets (and one is the winner). I choose one. The probability that one of the remaining 99 is the winner is 99/100. Then someone removes all but one of those 99 saying that they are all losers. Then it seems obvious that you'd want to switch. I don't have the exact math at my fingertips for this one but my intuition feels much better about it. sigh. BTW here's where I found this.

05-25-2003, 10:51 AM	#3
Undertoad Radical Centrist Join Date: Jan 2001 Location: Cottage of Prussia Posts: 31,423	We did this one on the Cellar mk 2!

05-27-2003, 03:12 PM	#5
arz Hand-of-Kindness Extender Join Date: Sep 2002 Location: So Cal Posts: 137	Well, strangely enough in 5 runs of the applet I stayed with my original guess all five times and won the $ 4 of those 5... 5 runs is not statistically valid and yet the initial intuition answer is reinforced not disproved. Update: I ran the simulation 12 times; 6 times without switching and 6 times with switching. Result: I won 4 times out of 6 in either strategy.

05-27-2003, 03:32 PM	#6
Undertoad Radical Centrist Join Date: Jan 2001 Location: Cottage of Prussia Posts: 31,423	Applet schmapplet. I'm willing to make you a deal right now. You put down $100, and I'll put down $500. I'll lay out 52 cards, face-down, and I'll look at them while I lay them down. You have to pick the 3 of clubs. You pick your first card, but don't turn it over. I'll take away 50 other cards and show them to prove that they aren't the 3 of clubs. Now if you have the 3 of clubs, you win the pot, while if I have the 3 of clubs, I win the pot. Agreed?

06-13-2003, 01:15 PM	#13
vsp Syndrome of a Down Join Date: Jun 2001 Location: West Chester Posts: 1,367	And it has never made sense to me, for a simple reason: I pick Door #1. Monty opens Door #2, and Rosie O'Donnell is standing behind it, so obviously I didn't want that one. He offers me the choice of keeping Door #1, or switching to Door #3. What I do not, and probably will never understand is how the second choice is not _independent_ from the first one. Yes, Monty has information that I do not as to which door has the prize -- but the mere fact that I do not have that information makes it irrelevant to my choosing process. I don't know whether Monty is trying to help me or screw me over by making his offer, and I have no way of finding out short of making a blind choice; all I _do_ know is that between my first choice and my second, the situation has changed and my options are different. When I make my second choice, I am making an independent choice between two doors. One has a prize. One has a goat. A third door has a goat, but that door is _no longer an option_, so I don't grasp why it is figured into the odds when I choose again. (Really, if Monty always offers you the choice to switch, the first choice is irrelevant; he will always open a goat door, so you'll always have a final choice that narrows it down to two.) I've taken math courses through Calculus 3. I've taken Probability and Statistics courses. I've read lots of explanations of the "real" odds for each door... and I _just don't get it._

06-13-2003, 01:21 PM	#14
wolf lobber of scimitars Join Date: Jul 2001 Location: Phila Burbs Posts: 20,774	I don't get it either, but such a thing does not prey upon my mind. I, personally, might be excited to win the family of goats. My enjoyment of the show was to see what goofy costumes the contestants were wearing, and I also loved the segment where Monty would look around the audience for an assortment of bizarre objects that women might have in their purses ... "I'll give you $50 for a hot dog ..." __________________ wolf eht htiw og "Conspiracies are the norm, not the exception." --G. Edward Griffin The Creature from Jekyll Island High Priestess of the Church of the Whale Penis

06-13-2003, 02:44 PM	#15
russotto Professor Join Date: Jan 2001 Posts: 1,788	In the two player version they ran sometimes, it was best to _NOT_ switch. The way that one worked, two contestants would each pick a door. Monty would then open a losing door which belonged to one of the contestants, and offer the other contestant the chance to switch to the remaining door.

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