In a comment on my post Top Secret Tips For Winning Game Shows, Marelisa the abundance queen reminded me of the Monty Hall paradox, which deserves a post of its own.
This problem became famous in 1990 when Marilyn vos Savant wrote about it in Parade magazine. This is how she stated it:
“Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to switch your choice?”
She said that you should switch, because switching will give you a 2/3 chance of winning the car, compared to 1/3 if you stick with door #1.
Not exactly intuitive, is it?
She received more than 10,000 letters from people (including 1,000 Ph.D.s) saying that she was wrong (and presumably saying the odds were 50/50 either way). She also got a letter from me, saying something different (we’ll come back to this).
Russian roulette
Here’s another way of looking at it. Let’s say you’re playing Russian roulette with one other person, using a gun with 6 chambers and 1 bullet. You spin the cylinder and you’re about to pull the trigger to fire chamber #1, which has a 1/6 chance of having a bullet.
But before you do, the other guy asks for the gun, saying he’ll fire four of the other chambers first. So he takes the gun and pulls the trigger 4 times, firing cylinders #2, #3, #4, and #5. Luckily for him, they were all blanks. He gives the gun back to you, and asks if you want to stick with your original cylinder #1, or switch to cylinder #6.
Because he didn’t know where the bullet was, it doesn’t matter if you switch. It’s a 50/50 chance either way.
But let’s back up and try something else. You’re about to fire chamber #1 when the other guy asks for the gun, saying he’ll fire four of the other chambers first. He takes the gun, but this time, he swings out the cylinder to see where the bullet is. Then he fires chambers #2 and #3, skips chamber #4, and fires chambers #5 and #6, all of which are blanks. He then asks if you want to stick with cylinder #1, or switch to cylinder #4.
Does it matter if you switch now? You bet! He skipped chamber #4 for a reason. And if the bullet was in any of the other chambers, you can bet that he would have skipped those instead. There’s a 1/6 chance that the bullet was in chamber #1 and he just skipped a random chamber to mess with you. But there’s a 5/6 chance that the bullet was not in chamber #1, and he specifically fired all the remaining blank chambers, leaving just the bullet.
Just like you should stick with chamber #1 to avoid the bullet, you should switch to another door to win the car.
The real answer
However, one of the reasons there was so much debate about this problem is that it’s ambiguous, and that’s what I said when I wrote to her. We just don’t know enough about the host’s reasoning for picking the door to open. For example:
- If the host had decided that he was going to open door #3 no matter what, and it just happened to be a goat, your odds are 50/50 with either door.
- If the host wanted to be nice and only offer you the chance to switch if your original choice was wrong, and only open a door containing a goat, you’d have a 100% chance of winning by switching.
- If the host wanted to be a jerk and only offer you the chance to switch if your original choice was right, you’d have a 100% chance of losing by switching.
To spell out Marilyn’s assumptions, the problem would read like this:
“Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. The host, who knows what’s behind the doors, has decided in advance that after you pick a door, he’ll open another door and give you the choice of switching to the remaining door. He’ll decide which door to open using this logic: if you pick the door with the car, he’ll open another door at random; if you pick a door with a goat, he’ll open the other door with a goat.
You pick door No. 1, and the host opens door No. 3, revealing a goat. He then says to you, ‘Do you want to pick door No. 2?’ Furthermore, he explains his logic in choosing to open door No. 3, and you know he’s telling the truth. Is it to your advantage to switch your choice?”
It kind of ruins the fun to spell out everything like that, but you can’t talk about who’s right and who’s wrong if the problem is ambiguous! When the problem is stated this way, the answer is that you should switch, because door #2 has a 2/3 chance of having the car. The way Marilyn stated it, it’s too ambiguous to answer unless you make some assumptions.
BTW, did you assume that the car is better than a goat? In a 1999 auction, someone paid $80,000 for a full-blood adult South African Boer goat. So maybe you shouldn’t switch after all!
Wow Hunter,
After reading that, I got confused. Is there a right answer? Other than, duck if someone has a gun pointed at you.
Umm, I still don’t get it. Will you explain again, please?
Or maybe it’s easier to hire a “psychic”.
Yeah, this is confusing, isn’t it? (And I see now that I forgot to explicitly say the answer, so I added a couple sentences near the end.) The answer is that since the question isn’t clear, you can’t answer it. But you should switch if you’re making the top secret assumptions that Marilyn made, which you’ll only know if you hire a psychic who specializes in game shows.
Oh dear lord, and I have even had my first morning coffee. I never did like probability, and I’m a horrible bridge and poker player. One year the Boyz at work convinced me to enter the Football pool. I know NOTHING about football. Couldn’t care less actually. Needless to say, they were furious when I won. They were beyond furious when I told them how I won. I simply closed my eyes, and stabbed my finger at the list.
There’s no way the host wouldn’t know which door really had the car, otherwise he could accidentally show you the one with the car behind it instead of a goat. I can see how it really boils down to your perception of the host’s intentions: to help you win or to help you lose? Why might he do one or other? Pacing and tension, perhaps.
Michael Martines last blog post..Remarkablogger Manifesto: What is Non-Negotiable?
Panther,
Maybe you can be the psychic who specializes in gambling
Akemi – Yes to Mes last blog post..A Year Without Paychecks, Part 2
Hunter: I don’t think that you can take into account the intentions of the game show host because you have no way of knowing what his intentions are. I think it just boils down to mathematics that when you choose door 1 there’s 2/3 probability that the car is behind door 2 or 3, and when the game show host opens door 3 and you see that the car is not there, there is now a 2/3 probability that the car is behind door 2. Because 2/3 probability are better odds than 1/3 probability, you choose door 2. But then again, I know that there was a lot of controversy when Marilyn vos Savant printed this, so . . .
You were one of the people who wrote to her?
Oh, and thank you for the link and the “abundance queen” bit
Marelisas last blog post..Success – On Your Own Terms
This post gave me a headache.
“Abundance queen” – awesome!
Vered – MomGrinds last blog post..Wordless Wednesday: Mona Lisa, Enhanced
@ Urban Panther, I can see how that would infuriate them. Especially if you decided to have some fun with it and keep laughing at them every time they lost. “You bet on San Diego? Are you out of your mind? My finger was clearly guided to Carolina. I guess you just don’t know as much about football as I do!”
@ Michael, I agree that on a real game show, he wouldn’t just decide to open the door with the car behind it. (Though he could have said, “Do you want me to open door #3? It might help you out, or it might have the car!”) But then, sometimes logic puzzles aren’t meant to describe a realistic scenario.
And when we get down to thinking about the host’s intentions as if it were a real show, it gets really complicated. I don’t think Marilyn meant to have us consider any mind games, but unfortunately we have to, considering the way the problem was worded.
@ Akemi, I noticed that you don’t offer a game show reading service, so I think it would be great if Panther could complement your offerings.
@ Marelisa, but without making assumptions about the host’s intentions, you can’t say that there’s a 2/3 chance. Consider this: what’s wrong with the following logic?
“There’s a 2/3 probability that the car is behind door 1 or 3. When the host opens door 3 and you see that the car is not there, there is now a 2/3 probability that the car is behind door 1.”
What you said is correct, but only if you know that the host was going to make the offer to switch in any case, and he was going to open a door with a goat in any case (as opposed to opening a door at random, or always opening door 3, for example). Some people think these assumptions are obvious, like the assumption that the car is a better prize than the goats, but I don’t think so.
Yeah, I wrote to her. They didn’t have email then, so it was snail mail.
I wanted to use good anchor text and branding for your link, so I thought abundance queen would work!
@ Vered, yeah, I have a pretty big headache too! I ‘d like to know what anchor text everyone prefers, but I guess it’s OK if I just make up things like “abundance queen!”
I’m reading this post in the morning too. I went like “huh”…??? I think I’d better go back to bed. I’m still not very alert. I don’t want to walk into a street lamp on my way out while trying to think about your post.
Evelyn Lims last blog post..Attract Our Travel Dreams
@ Evelyn, I guess I should have a note saying “WARNING: Do not read in the morning before coffee!” Be careful about walking into street lamps today.
I (heart) Monty Hall.
Sorry, but you lost me. It’s late though, so maybe I’ll come back tomorrow AM after I drink my coffee.
@ Linda, it’s a confusing problem, which is why 10,000 people (1,000 of them Ph.D.s) wrote in to tell Marilyn she was wrong. If that many people wrote snail mail letters, how many more thought she was wrong and didn’t bother to write, and how many more were confused? We will return to “easier to understand” programming shortly.