Monty Hall problem on MythBusters

Cafe Society
81

How are they going to determine which of three doors you picked when your choices were left or right? Effectively, you weren’t given the choice of the middle door.

I know people have said this already but one more time. They hide a prize behind one door out of three choices. They let you randomly pick one door. The host then offers you a choice - take what’s behind the one door you already picked or trade it for what’s behind both of the other two doors.

Hopefully you can see that two chances are better than one chance. And that’s what the classic set-up does. Putting aside the bit about opening an empty door, it’s letting you trade the contents of one door for the contents of two doors.

82

If you know the host knows which door has the prize and you know the host will choose which door to open in the manner proscribed by the Monty Hall problem, then yes, the door Joe doesn’t pick has a higher chance of holding the prize.

The problem for a subsequent contestant is that he doesn’t know which door Joe picked.

If you are told which door Joe picked, and you know why the host picked the door he did, then you increase your chances of winning from 50% to 66%. This is not like quantum physics. This is you learning more information about the situation, so you can better refine your guestimate of where the prize is.

83

I’ll try to explain this one more time.

You are actually given a choice of one door or two doors. If you pick one door, you win if the prize is behind it. If you pick two doors, you win if the prize is behind either door. Which choice would you make?

Here’s what happens. You name a door. If you want to choose one door, that’s the door. If you want to choose two doors, that’s the door you aren’t taking. After you name a door, you are given the choice between one door or two doors. Since you have no idea which door the prize is behind, it doesn’t matter which door you name to start with.

It doesn’t matter that you are shown an empty door out of the two door set before making your decision whether to go with two doors or one door. That’s a red herring.

It’s that simple. Which has better chances of winning? Two doors or one door?

Let’s examine in a little more detail.

First, you name a door. The question implies you are choosing the door you think has the prize behind it. But really, you haven’t decided you want that door yet. You can still say that’s the door you don’t want.

Second, you get to choose between the one door you named, or the other two doors. Remember, you win if the prize is behind either door if you go with the two door option.

What about being shown a door with no prize behind it? It doesn’t matter, if you choose the two door option, you win if the prize is behind either door. That’s just a distraction.

It’s pretty easy to see if you choose one door, you will win 1 our of 3 times. If you choose the two doors, you will win 2 out of 3 times.

84

Bolding mine.
Although I understand the problem, I did this for fun anyways and you only won 6/15 times (if I graded you correctly…I’m rather tired!). I didn’t do your second set of data.

The flaw that became apparent to me is that as I was setting up my “random” set of solutions in Step 1, I was thinking about it… I would have had to find a random number/letter generator because in my attempt to come up with a random pattern I wasn’t. Does that make sense?

You and I often made the same choices in our attempt to be random. There’s probably a psychological reasoning behind this. You know how you start to panic when you take a multiple-choice exam and you realize you just filled in the 'A" bubble on the scan-tron three times in a row, but the next questions right answer seems to be A… we convince ourselves it can’t be right, even if it is.

Presumably the location of the prize is truly randomized in the Monty Hall Problem.

Data:

Yours - Mine

  1. C B
  2. B A
    **3. A A
  3. C C**
  4. A C
    **6. B B
  5. B B
  6. A A**
  7. C B
    **10. A A
  8. C C
  9. B B**
  10. C A
    **14. B B
  11. A A**
    With a truly random set, and/or many more example runs, the probabilities would likely converge to the expected 1/3 and 2/3, but some weird sort of mind-reading happened in this case.

Sorry you lost. I’ll have the goat delivered in 4-6 weeks. You’ll be taxed on the value of the goat…:wink:

85

Pure math puzzles and real-world puzzles are distinct.
What makes Monty Hall problem (slightly) interesting is not to specify that host will open an empty box. That fact should be obvious to anyone with common sense about the real world, who realizes Monty wants viewers to be in suspense enough to not switch channels during commercial break.

“Two choices, so the chance is 1 in 2.” I’ve listened to several intelligent people who believe this, including a clever circuit designer/engineering manager.

86

I know that’s a rhetorical question, but here’s my offer: two cents… deal or no deal? :wink: Because without the banker being the Bad Guy, there’d be no “conflict” and it would be just some goober up on stage picking numbers. Unlike other games, there are no other contestants to play against, no angry mob, no Whammies.

Interestingly, on the variation of Deal or No Deal where the potential contestants are onstage and randomly assigned their cases, the chosen player has the option to switch before they start playing. A fair number of them do, unlike the regular version where they never want to switch at the end. It’s like because they didn’t select it, they don’t have that “I made this choice and I’m sticking to it” mentality.

87

No, you misread my story.. the door by the person was chosen before joe chose, without ever meeting joe. You said “left or right” before joe chose, before you went in the room.. then you go in, and there are two doors, you get the one corresponding to the choice you made.

88

Hey , read my scenario! I said you picked left or right BEFORE.. the door that was opened by the other host was removed.. whey you walK in you don’t even know there were ever 3 doors!

89

Suppose I chose left, and this happened as you described. What would have happened if I chose right? We can’t assign a probability here because the rules of the game aren’t specified.

90

Why is this confusing?

JOE may think he’s got a 50/50 shot, but he’s wrong. He’s not in possession of all the facts (also, you need to specify that the prize can be distributed to any of the doors with equal probability).

Ever roll dice? Let’s say I roll a die and tell you I got a 5. I ask you what the probability was (before I rolled) that I get a 5. You tell me 1/6.

Now, I tell you, “Aha! It was actually a 20-sided die and not a 6-sided die. The true probability was 1/20”.

Your probability figure is incorrect because you are not in possession of all the facts.

The same is true for JOE. But WE, knowing all the rules and procedures, have a better idea of the actual probability.

The point is not to determine what probability Joe thinks he has. It’s to determine the true probability, which can be significantly different than what the contestant can deduce, based on the limited information provided.

In the real Monty Hall problem, the contestant is given all the facts ahead of time, as opposed to this scenario, where information is hidden from the contestant. That makes all the difference in the world (at least from the player’s perspective, if not the audience).

91

Wait a sec. How do you have no information? 24 cases have been opened, by you, randomly (or superstitiously, but what difference does that make).

Why don’t your chances of getting the better prize go from 1/26 to 25/26 by switching?

92

It’s due to the lack of knowledge by the opener of the cases. In the classic Monty Hall, as explained above, the key is that Monty knows where the prize is. The door with the prize will never be opened, and that fact is what makes the probabilities inequal.

If, in Deal or No Deal, you had Howie Mandel opening cases for you, and he knew where the million dollars was, and he intentionally never opened the million dollar case, then yes, you’d gain a ton of equity by switching at the end.

But that’s not the case. You pick them, and you have no knowledge. Therefore your chance to gain equity goes out the window by the simple fact that you can hit on the big money prizes in a way that Monty Hall never will.

93

The Monty Hall problem is a pretty classic case of people who think they’re clever trying to lord it over other people when they’ve actually got it wrong.

Yes, if Monty always gives you the option to switch, you’re better off switching, since you only had a 1/3 chance of being right in the first place. That’s just basic probability. But that’s not the game. Monty’s not compelled to give you the choice. Sometimes he does, sometimes he doesn’t. So if you are given the choice, you’re really back at 50/50. No reason to switch.

But some people find it fun to go to cocktail parties and find someone who’s not familiar with the problem and laugh at them when they they give the naive “wrong” answer that there’s no benefit to switching. The victim may be right for the wrong reasons, but if the asker doesn’t explain that they’re changing the rules of the game, or (more often the case) doesn’t even realize themselves that they’re doing so, they’re totally wrong.

94

If Monty only offers the choice sometimes, the advantage to switching depends on how much information his actions give you about where the prize is. It’s not necessarily 50/50; for instance, if Monty gives you the choice with 2/3 probability regardless of whether you’ve chosen correctly, switching is just as advantageous as it is when Monty always gives you the choice.

95

And what would the point of that be? Monty’s goal is to make the odds as close to 50/50 as possible, so we can assume any action he takes is to serve that goal.

96

Says who?

97

No they aren’t. Monty’s goal is to maximize his show’s ratings. If he figures he can best achieve that by stacking the odds against a contestant, then he will. If he figures he can best achieve that by leading the contestant straight into the grand prize, then he’ll do that instead.

But “The Monty Hall Problem” has nothing to do with the actual game show. It’s a hypothetical case designed to show how our intuitions about probability can diverge widely from actual probability.

98

Actually it’s not our intuitions about probability. The problem itself is described in a misleading manner, leaving people with the correct impression about probability, but unfortunately for the wrong problem. If you deconstruct the problem, it is a choice between 1 door or 2 doors. Once informed of that, everybody correctly understands.

99

How is it misleading?

100

Read post #83. The choice of the first door is not really the choice at all, but the player feels like they’re trying to pick the prize. Revealing an empty door is a distraction. It’s actually what makes the choice between 2 doors or 1. Instead people feel like it’s reducing the problem, but actually it isn’t. The real choice doesn’t come until the switch. That’s when you decide whether you want what’s behind 2 doors or 1. Obviously 2 doors gives you better results.