Monty Hall problem on MythBusters

Cafe Society
1

The recent episode of MythBusters featured four fan-submitted myths, one of which was perennial Dope fave The Monty Hall problem.

I was impressed with the way they handled it. They divided it into two problems: first, that there is a strong psychological impulse to stick with your initial choice (assuming you aren’t aware of the true probabilities) and second that switching is always a better bet (which it is.)

For the first, they had a horde of 20 volunteers play “pick a door” on a genuine three-door game show stage, with Jamie randomly resetting the prize door and Adam being Monty Hall and offering the switch. Every one of the 20 chose to stick!

For the second, they built a very clever door-picking-game simulator in which Adam and Jamie played 100 games, with Jamie always sticking and Adam always switching. Naturally, the results indicated that switching was a vastly superior strategy.

2

My dad asked me the Monty Hall question when I was a kid and I got the answer right the first time. I used the same explanation then as they used. You only had a 1 in 3 chance of getting it the first time, so there’s a 2 in 3 chance with the other two doors. If you remove one of the other two doors, so it’s just one door, then it takes the full 2 in 3 odds on itself.

3

I didn’t understand it (I accepted it, but I didn’t understand it), until I realized that switching always reverses the outcome.

4

I’ve read of this many times before, and even the practical demonstration didn’t make sense to me. So I still don’t understand the maths behind it.

5

Lord Feldon’s insight is a really good one. Switching always reverses the outcome. If you picked the winner, then obviously switching will cause you to lose. If you picked a loser, the other loser will be eliminated when Monty opens the door, so switching guarantees that you will win. The probability of picking a loser is 2/3, so the “always switch” strategy gives you a 2/3 chance of winning.

6

There are three decisions being made:

Which door does the contestant pick?
Which door does the host open?
Does the contestant switch doors?

The argument that switching doors is the best tactic relies entirely upon the host’s decision not being random. In other words, you’re not playing Three-card Monte, you’re playing poker.

7

That’s a nice way of explaining it.

8

For me, at least, the the problem is easier to visualize if it’s scaled up: Imagine a really rubbish game show where instead of three doors, there are 100 doors. You choose one, and then Monty says: “I’ll make this easy for you, because this is the worst game show ever. I’ll now open 98 of the remaining doors, where there is no prize. Only one of them remains closed. You can keep your door, or switch to my door instead.”

Now, think. There are really only two sets of doors: The one door you picked, and the set of doors you didn’t pick. With 100 doors instead of three, your chance of picking the right door on your first pick was just one percent. The chance of the prize being behind a door *somewhere *in the set of the doors you didn’t pick, is 99 percent.

When Monty asks if you want to switch, he’s really asking: “Do you want to keep the door that you initially picked, or switch to the full set of doors that you didn’t? I’ve already opened 98 of those doors, so if the prize was anywhere in this set from the get go, and you switch, you’ll win, guaranteed.”

In other words, if you picked the right door to begin with, you win if you keep your door. If you picked the wrong door to begin with, you’ll win if you switch. But your chances of picking the right one at the start was only one percent!

In this case, the answer should be a no-brainer. Doh, switch every time! You’ll get a 99 percent chance of winning!

Now, in the actual Monty Hall problem, the situation is exactly the same, just with different numbers. Your chance of picking the right door to begin with was 1/3. The chance of picking the wrong one was 2/3. Switching gives you a 2/3 chance of winning.

9

But surely what happens at the start is irrelevant. The final decision you make is 1:2 because those are your only options.

10

And they ask me why I drink. :stuck_out_tongue:

11

No. If you had no information with which to make your decision, it would be a fifty-fifty chance. But you do have information: you know that the host knows which door is the right door, and you know that the host will act in a known, non-random manner.

12

Imagine 100 doors, you pick one, then the host tells you it’s not these 98 doors. Do you change? Of course! Because now you’re really choosing 99 doors all at once versus the 1 in 100 chance you had before. But if you walked into the studio and had no idea was just happened, picking either closed doors is just as likely.

Mythbusters is really stretching, tell me they at least blew up some doors at the end. Modern Marvels is losing it too, I expect an episode on the paperclip soon.

13

[hijack] Actually, please tell me they didn’t. There have been way too many shows lately where they’ve just gone: “Oh, we couldn’t think of any good myths this week, but that’s OK - we’ll just blow something up or shoot a really big gun, and surely no one will notice.” [/hijack]

14

One thing I noticed was that Jamie said there was no point in telling us how many of the first group won or lost because they all picked to stay so their results wouldn’t compare the two strategies. But he was wrong. It’s a game with only two outcomes - so everyone who lost by staying would have won by switching (and vice versa if that had happened). So if we knew, for example, that seven people stayed and won and thirteen people stayed and lost, then we’d know that switching would have won in those other thirteen cases.

15

Don’t worry - Monty was only 1/4 of the episode. The other parts involved grenades, gangsta-style shootouts, and a hatchback loaded with two tons of building supplies.

16

That is not inherently true. It could also mean the host is trying to trick you into moving away from the correct answer. Like I said, it’s not about probabilities, it’s about knowing how the host responds to information he can see that you can’t, to work out what he knows.

17

That’s the thing, though. The way the problem is usually set up (at least, every time I’ve ever seen it) is that you know there are three doors (one having the prize behind it), and you know that the host knows where the prize is, and you know that the host will always open a door with no prize behind it after you make your initial selection. Of course, this may be different than what happened on the actual TV show, but the “Monty Hall problem” is not meant to exactly copy the circumstances of the TV show. Rather, it’s to set up a specific mathematical thought experiment.

I’ll also mention that there was a short scene on the Mythbusters episode with Adam establishing precisely those conditions I described above. So in other words, they were testing the standard set-up for the problem. If those conditions are not expressly given when posing the problem, then it will change the answer.

18

Rules of the following game:

  1. I think of a number between 1 and 100.
  2. I ask you to guess a number between 1 and 100.
  3. I tell you that I’m going to rule out 98 numbers, one of which is the number you chose. If you chose correctly in the first place, then I’ll choose another number randomly not to rule out. If you chose incorrectly in the first place, then I won’t rule out the correct number.
  4. I give you the opportunity to switch to the other number, or stick with yours.
  5. I reveal whether your final choice is the number I thought of in step 1.

Let’s try it out; I’ll fill in your guess more-or-less randomly in step 2.

  1. I’ve got my number, but I’m not telling you what it is.
  2. You guess that it’s 47.
  3. I tell you that it’s not 1-46, 48-72, or 74-100. It’s either 47 or 73.
  4. You can now switch your guess to 73, or you can stick with 47.
  5. If you stuck with 47, you’d be wrong. If you switched to 73, you’d be right.

Step 3 is important. I’m never going to rule out the correct number, any more than Monty is ever going to open the door with the prize and say, “Oops, too bad!” I’m ALWAYS going to rule out an incorrect choice. That move on my part is what skews the odds so heavily in favor of switching for you.

If you’re still not convinced, play the game I described above with a friend. If you follow the rules correctly, if you switch every time, you’ll win 99 times out of 100. If you stick with your original number every time, you’ll win 1 time out of 100.

19

The fascinating part of this ep. isn’t the math involved, but the psychology. I expected beforehand that, yeah, the majority would stick with their choice, but maybe 1/3rd would switch, if for no other reason than to be contrary. I was absolutely blown away when not a single solitary soul changed his/her mind; they invariably gave excuses like “I sticking with my gut”. Even if I didn’t know the math involved going in, I know from playing boardgames and such that your strategy should be random so as to not leave yourself vulnerable to your opponent correctly gauging your tendencies (which in the case of the original Monty [del]Haul[/del] Hall game show he would most certainly do if given the chance). In other words I would expect the host to expect me to stick, like most everyone else would, so I would thus confound his strategy (assuming there was one) by switching.

20

Martian Bigfoot and the Left Hand of Dorkness explain the maths behind the “Monty Hall” problem perfectly. If you’re still having problems trying to understand it, you’ve got to try and “think big”.

Instead of 100 doors, try visualizing 1000. Or a thousand million. What’s one thousand million times infinity? They’ll try and tell you that there’s no number larger than infinity, but they’re wrong.

So imagine three infinite sets of infinite doors. And behind each door is one hundred thousand years, one hundred thousand lies, one hundred thousand eyes. Each set of doors takes an infinite amount of time to open. When the eyes open, you must avert your gaze unless you burst into flame.

And the flames of justice burn bright! They will tell you that they are trying to “bust” or dispell the “myth”. Was Enoch a myth when he saw the “burning wheel”? The wheel that always revolves at an “infinite” amount of time? They will try and “tell” you “no”.

Seven times seven angels dance. Can you read the signs? One of them stops your car as you drive. She gives you a choice: to continue down the road you are on, or follow one of the new paths that he illuminates. But only one! While you think, the “others” try to interfere. They are stopped and turned away.

You stop time and return. The maths and lies and myths are left behind, like childish children’s childish crimes. The angels breathe, and kiss you a final gift. “We will close all the doors”, they sing and sigh and sign, “all but the won you dearly wish to take”.

To choose and win? Your final fate. The mountain looms ahead. The mountain hall and hall of kings, the King that Enoch tried to spell, the myth that larger man must surely dread.

Pretty simple, really.