# Talk:The Monty Hall Problem

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Layout issues

There are a couple places where the break between paragraphs are extra large. Is this intentional? (Abram, 7/9)

Yeah, if I make them smaller then all the pictures get bumped and it looks weird 7/21 Becky

Monty and Monkeys section

For whatever reason, my eyes want to glaze over the Monty and Monkeys section. Is there any way to add another image or two in there to break up the text? (Anna 7/7)

I like your new picture, but for some reason this is the one section that I still feel is the weakest. I'm not sure of what changes to make, but I think some small tweaks could be helpful. Can you get Iris and Xingda to look at it? (Anna, 7/15)
Sure... i'll recruit them..
Did they have anything to say? (Abram, 7/19)
I'm not Iris or Xingda, but I like this section. I think it's really interesting and clearly presented. Here are a couple of small things that might help, though I doubt any of them are actually what Anna is worried about.
• In one place you refer to the colors as being roughly equally favored, and later you say Keith Chen assumes an order of preference. What's the deal?
I tried to make it more clear in the section.. basically Chen is saying that their assumption that the monkeys like all three colors equally is wrong. He's saying that we can't ignore a slight preference that the monkeys might have. If he's right that there is a slight difference between the three colors, then the study didn't prove anything at all. 7/16
If you want to, you could emphasize more strongly that even a very small difference in preference will lead to this very large 2/3 vs. 1/3 differential. I'm just a bit worried that right now when people read the sentence "If this is the case, then the monkey's choice of yellow over blue wasn't arbitrary," will say, "Well, of course not, you just said there's a small difference in preference." (Abram, 7/19)
• John Tierney is a bit of a sensationalist. Can you find anyone else talking about this study? Don't worry about it if you don't want to.
No, I looked. He's the only one who has covered it so far.... 7/16 Becky
OK, you could add this is a direction for future research if you'd like. (Abram, 7/19)
• You might want to add a bit of vertical space between the top half of the principal image for this section and the text "Cases where the monkey prefers... are highlighted above".
Did this. 7/16 Becky
Looks good. (Abram, 7/19)
• In the bottom half of the picture, it's not totally clear that you are trying to say that the color of the red/blue M & M is the winner in a red v. blue showdown.
Ok, i changed this picture a little and added more explanation 7/16 Becky
Looks good. (Abram, 7/19)
(Abram, 7/15)

Cite the 1956 study

Including a reference would be a good idea, but at the least say who the study was conducted by and what its name was. I found myself really wanting to look up this study, but didn't want to track down the reference myself. This, and the issue with Bayes section I added on 7/9, are the only remaining comments I care all that much about. (Abram, 7/9)

I put the name of the study into the text, and then I put the full information about it in the references section. Do you think this way works? Becky 7/9
Yeah, that's great. We just need to work on footnoting in general. That's a topic for Friday's meeting, though. (Abram, 7/15)
Maybe include a link to the footnote? (Abram, 7/19)

The Extreme Case

The Extreme Case section is a brilliant, intuitive explanation of the solution. Very well presented.

The nitty gritty itty bitty details

In the Bayes section, I'd edit this sentence: "This is essentially asking if the probability that the car is behind one unopened door the same as the probability that the car is behind the other unopened door. " I'd definitely get rid of your first few words in that sentence. I might actually add a sentence saying "$P(V_a\|O_b)$ is the probability that the the car is door A given that door be as been opened..." just to help clarify what you're talking about. You might also want to draw your picture again labeling doors a, b and c, with door "b" open. (Anna 7/7)

Hey Anna, I changed that sentence and expanded the explanation to readdress the meaning of $P(V_a\|O_b)$ etc... I think it's much more clear now. Rebecca 14:54, 8 July 2010 (UTC)
I liked your changes, though I added in a strategic line break to make it a tiny bit more readable. I also don't see a need for a picture, now. (Anna 7/14)

Including a picture in this section, as Anna suggests, sounds like it might be a good idea. The only thing to be careful of is that the whole approach to the Bayesian solution is wildly different from the "every scenario" solution, and you don't want a similar looking picture to fool readers into thinking there is similar reasoning going on. (Abram, 7/8).

Does this picture work? I decided not to add the pic with door B open because I'm afraid people might be confused by skipping to the step where monty has already revealed a door. Showing door B open without showing that the contestant already selected a door might be confusing, but showing the contestant selecting a door will make the pic look too similar to the decision trees in the first section... So i couldn't really find another way to get around both those issues. Rebecca 14:54, 8 July 2010 (UTC)
It seems like everybody is happy with the current layout. (Abram, 7/15)

Two other thoughts on this issue. First, in the presentation of the non-Bayes solution, you begin by saying, "We first imagine that the car is hidden behind door 1", and finish by saying that the reasoning is the same even if the car had been hidden behind a different door. Can you do a similar thing for the choice of Monty opening door B for this presentation.

Ok, I changed this. Rebecca 14:54, 8 July 2010 (UTC)
Oh wait, there's a problem. The end of the Bayes Theorem section says we can generalize the solution because the car is equally likely to be hidden between any of the three doors. But we started with Monty opening door B as a given, not with the car being behind a particular door as a given, and with this given, the point is that the car is *not* equally likely to be behind any of the three doors. In fact, the generalization should be that Monty is equally likely to open any of the three doors, which is a weird way to think about it, but actually the correct way. Also, you might want to put a framing sentence at the beginning, like, "We begin by looking at a scenario that leads to Monty opening door B." (Abram, 7/9)
I tried to change this. I think the wording is better, but you might want to see if you think it works. I put that framing sentence in the beginning, and changed the part at the end. Becky 7/9
I think your change works (Anna 7/14)
I think the change looks good also. Theoretically, a reader could claim that Monty isn't equally likely to pick any door, because he can't pick the door with the car, and therefore the last sentence of this section doesn't generalize. Addressing the flaw in this reasoning, though, is a lot of work, and probably outside the scope of the page. Do you agree? (Abram, 7/15)
Yeah, sounds like we can just leave it for now at least. 7/16 Becky

Do a little bit more guiding in the initial presentation of the solution.

Before you go through the whole solution process, it might be helpful to tell people what solution you are about to come up with, and how you are going to do it. E.g. "It is to the contestant's advantage to switch: the probability of winning if the contestant doesn't switch is 1/3, but if the contestant switches, the probability becomes 2/3. To see why this is true, we examine each possible scenario below". Then start the solution. (Abram, 6/25)

Ok, I added this. Rebecca 19:12, 25 June 2010 (UTC)
This has looked great for a while. (Abram, 7/15)

The Bayes' Theorem Section is actually correct!

We just have to replace "the door opened by Monty Hall be called b..." with "the other doors are called B and C." (Abram, 6/18)

Changed this Rebecca 14:34, 24 June 2010 (UTC)
Looks good. (Abram, 6/25)

Talk more about what people tend to mess up

You give a really good account of how often people mess the question up. Somewhere, it would be great to say what the wrong reasoning is that people tend to employ.

We talk about including this description of the conventional reasoning *before* you give the correct reasoning. Then *after* you give the correct reasoning you can have a brief section that examines the fact that sometimes informal reasoning seems totally valid, but gives a different answer than examining every possible outcome. (Abram, 6/18)

I tried to do this. The section is actually organized a little differently, but you might think it's OK anyway. Rebecca 14:34, 24 June 2010 (UTC)
Yeah, nicely done. The only reason I would suggest considering introducing the wrong reasoning *first* is that presenting innocent-seeming reasoning and then saying, "Gotcha! That answer isn't actually right!" gives people a first-hand experience of how deceptive this problem is *before* they have to go through a somewhat arduous, less interesting derivation. (Abram, 6/25)
I tried to do this. Hopefully it's an improvement
Really nice job with this. The only slightly confusing sentence is "The critical fact is that the Monty does not randomly choose a door to open, so they do have a reason to prefer a certain door." Maybe something like, "The critical fact is that Monty's choice of which door to open is not random, so when he opens it door, it gives the contestant new information." You also might want to move the following sentence to the next paragraph (assuming the alien analogy is Marilyn's). (Abram, 7/8)
Ok, I used your wording and moved that sentence... Looks better now. Becky 7/9
Looks great. (Abram, 7/16)

Explain the drawing a little more clearly.

Add text and arrows to explain what's going on in each step and to show the sequence of events. (Abram 6/18)

Changed this Rebecca 14:34, 24 June 2010 (UTC)
Great. It's really helpful that the first picture has the caption, "Monty randomly picks which door to open". Can you put similar captions for the two scenarios to the right. (Abram, 6/25)
Yep sure, done. Rebecca 19:12, 25 June 2010 (UTC)
Looks good. I'm starting to think that the arrows indicating the "stay" or "switch" are a bit confusing. Also, I wonder if the revealed prize should be an open door with a circle around it or something, rather than a closed door. It's not that confusing, though, and maybe not worth the trouble. (Abram, 7/8)
Maybe we can talk about this when you come in... I see what you mean, but if I show the doors with the cars open then I feel like I have to show the doors with goats open when the contestant wins a goat, which means complicating the bottom of the image a lot. It seems like this might be unnecessary and just cause even more jumble around the bottom of the picture, where the general idea can still be gleaned from the big WIN LOSE LOSE signs at the end. I actually just read the comment at the bottom saying you weren't that worried about the picture. I'll probably just leave it then. Becky 7/9
I think everything you are saying is right. What about making the squiggly arrow connect hand to hand instead of door to hand? Or not, if you want to be done with it. (Abram, 7/15)
I changed the arrows and also a bit of the description for the last step of the pictures because I wanted to take out the big words STAY and SWITCH that I had written three times on each picture. I felt like they were seriously cluttering things. 7/15 Becky
Looks good. (Abram, 7/19)

I found two typos

• "To why this is true"
• Although Chen agrees that the study may have still discovered useful information about preferences, but he doesn't believe it has been measured correctly yet.
Changed these. 7/16 Becky
Looks good. (Abram, 7/19)

(Abram, 7/15)