# The Monty Hall Problem

I thought about titling this post, “In Which My Readers Grow Bored With Me.” But, I’ve been thinking about this all day, so I’ve decided to press on. I have simplified the argument here to bare essentials, though, so I hope you’ll excuse the lack of nuance. It’s a blog post, not a journal article.

In an interesting post on statistics, Tony mentions the Monty Hall Problem and the controversy surrounding the problem when it was written about by Marilyn vos Savant in her Parade column in 1990. This problem has always bugged me because it is often stated in a misleading fashion. In fact, I think that the statement of the problem in vos Savant’s column (quoted by Wikipedia) can’t quite be definitively answered.

Let’s look at vos Savant’s version of the problem:

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?

The fact that the host has a priori knowledge of the location of the car is extremely important (and is sometimes omitted from the statement of the problem). The implication here, which is frankly a bit of a stretch, is that of the two doors not selected by the contestant, the host will always open a door with a goat behind it. (There are, in fact, numerous other ways that the host might behave. Maybe sometimes the contestant is forced to open the door that she initially chose. Maybe sometimes the host actually opens the door with the car thereby showing you that you lost. Any possibility in which the host is not required to reveal a goat would lead to a different analysis.)

But, if you understand the basics of how the game is intended to work, then it turns out that switching doors is at least a weakly dominant action. Here’s the analysis usually presented: If I initially chose the door with the car behind it, which occurs 1/3 of the time, then the host will reveal one of the other two doors and, by switching, I will lose. If I initially didn’t choose the door with the car behind it, though, which occurs 2/3 of the time, then the host will be forced to open the door with the other goat behind it and I will win by switching. Thus, 1/3 of the time I win by staying put and 2/3 of the time I win by switching. Sounds pretty good. But, this does not answer the question posed. To answer the question posed, I don’t need to know the probability of a win when I use the strategy “switch no matter which door the host opens” – which is what we just computed. Instead, I need to compute P(Car Behind #1|Door #3 Opened).

In this case, though, you can’t actually compute P(Car Behind #1 | Door #3 Opened). The reason is that problem still doesn’t reveal the strategy used by the host, nor does it even clue you in that this might be relevant. Suppose we try to compute this probability: P(Car Behind #1 | Door #3 Opened) = P(Car Behind #1 AND Door #3 Opened)/P(Door #3 Opened) = P(Car Behind #1)P(Door #3 Opened|Car Behind #1)/P(Door #3 Opened) = (1/3)P(Door #3 Opened|Car Behind #1)/P(Door #3 Opened). And neither of the remaining probabilities in this expression can be computed. Specifically, the problem doesn’t tell you what the host will do in the case where you initially chose the door with the car. In that case, the host can open either door. Actually giving a definite answer as to whether or not it is strictly better to switch requires a more knowledge than the problem provides. Let me provide an example.

Suppose the host uses the following (admittedly contrived) rule: If the contestant chooses door #1 and this is the door with the car behind it, then always open door #3. Now, if door #3 gets opened, then the probability that the car is behind door #1 is 1/2! If the host is using this rule, then he opens door #3 if and only if the car is behind either door #1 or door #2 (which are, a priori, equally likely events). Hence, switching and staying put have the same probabilities of winning.

Now, it turns out that for any rule the host uses, switching will be at least as good as staying put. In fact, for every rule other than the one described in the previous paragraph, switching is strictly better than staying put. Thus, if you don’t know what the host is up to, you’d definitely ought to switch doors. But you can’t actually know whether or not it is to your advantage to switch in this specific case without more information about how the host will behave.

I agree with vos Savant in the sense that you can’t go wrong with switching doors in this particular problem. One can prove that the strategy “always switch” is strictly better than the strategy “always stay put.” Even better, the strategy “always switch” weakly dominates every other strategy regardless of what the host does. But, in certain pathological cases it may not be “to your advantage” to switch. I am crystal clear about the probability concepts involved. But, I feel strongly think that this problem has lots of unstated assumptions and nuances that are far from obvious. It isn’t just that it defies expectations – the problem is, in fact, under-specified.

# Pencils Down!

After noting that mackenab.com was up and running again, Andy sent me a link to relative new blog, Pencils Down, created by our mutual high school friend Tony. I promptly added it to my feed reader (the incomparable Google Reader) and my blogroll. And last night, I spent an hour or two after Charlie went to bed (when I should have been doing something else) reading through all of his posts so far. (His blog started in February. It currently has 183 posts. Some of them are substantial.)

I really like Tony’s blog. I won’t try to tell Tony’s life story here. He and I lost track of each other for long enough that I couldn’t do it justice if I tried. But, suffice to say, he hiked the Appalachian Trail last summer and moved to Maine. This fall, he starts back to college at the University of Southern Maine for a second bachelor’s degree with the ultimate goal of becoming a math teacher.

Two particular themes of his blog really resonated with me.

First, Tony insists in several posts on the importance of teaching probability and statistics in K-12, even if it means displacing some other topics. He rightfully points out (though not in these words) that these are topics which interact with people’s lives and in which our culture’s math illiteracy is most easily exploited. I agree completely. Probability is the primary mathematical tool that I use in my work. I mostly use random processes in my research, especially Markov chains and Markov decision processes, but in trying to teach these subjects I have been shocked at the poor preparation that many of our new graduate students have in probability. So, in a graduate course that is supposed to introduce students to random processes, we wind up trying to review all of probability up to that point.  And this is for graduate students in electrical and computer engineering.

This brings me to a slightly funny story. While I was at Cornell in graduate school, I found out that an acquaintance from high school was also there. (Tony and Andy might know him, in fact, but I will do him the courtesy of not naming him here.) We hadn’t been friends in high school, exactly, but I liked him well enough, and he was also in graduate school at Cornell studying, I believe, Chemistry. In any case, I ran into him on campus, and we agreed that we should get together. As it happened, Bela Fleck was coming to a nearby town, and he invited me to join him and another friend of his (also an acquaintance from our shared high school, as it happens) who would be in town for the concert. I’m not a Bela Fleck fan, exactly, but I like his music pretty well and so I agreed. In any case, on the drive to the concert I made an offhand comment about how important and misunderstood I think probability is. I probably overstated my case a bit, claiming, as I recall, that probability was the most the important area of mathematics. He disagreed, saying that differential equations were much more important and useful for modeling the world. I stood by my position, and he got quite angry. I can’t remember if he outright said that my position was idiotic, but he certainly strongly implied that it was deeply and obviously so. I sent him an email once or twice after the concert, but I never saw or heard from him again.

In any case, I think this is what our argument ultimately comes down to: If you are a physicist or a chemist trying to construct a precise mathematical model of the world, then differential equations may the most important tool in your toolbox. In my opinion, though, ordinary folks can get along just fine without them. A failure to understand the basics of probability, though, causes ordinary folk to make bad decisions on a regular basis.

The second theme of Tony’s that really resonates is his discussions of getting girls more excited about and interested in math. We have certainly struggled with the difficult issue of increasing the representation of women in our ECE department. It’s also the issue that has caused the most contention on this blog back in the deep archive. I’m glad that Tony and others in his community are thinking about it, though, because one thing that’s obvious is that no matter how hard we work to improve the status of women in ECE at Virginia Tech (and there’s lots that we need to do), this is a problem that often stretches back to girls’ early encounters with math and science.

I want to say more about a couple specific posts that Tony has made, but alas, I also have a lecture to prepare on mathematics in finite fields. (I’m teaching error control coding this semester, and all math for error control codes is traditionally done in finite fields. Since most students haven’t seen them before, we start there. In short, the first thing I try to teach my students is that, in the fields that we care about for this course, 1+1=0. They are surprisingly resistant.)

# So Very Close…

Well, I’ve tackled everything in the list below.  The photo up top is one of my own, from atop Diamond Head, looking back down Waikiki Beach towards downtown Honolulu.  The sidebar boxes are not design perfection, but I am fairly happy with how they came out.

And I’ve added a del.icio.us feed, which is something that I wanted to do with the old site but never got around to doing.  The feed will link to stuff that I think is interesting, but don’t necessarily have the time or inclination to write up in a full post.  It is also, at this very moment, empty.  That will change soon enough, I think.  Now, maybe I should do some real work!

# Theme Installed

As you can see, we now have a new theme. I basically like it (otherwise I wouldn’t have chosen it, I guess), but there are quite a few changes that I want to make. Obviously, I’d rather not have the Comments link in German.  (Done.) I’d like to change the weird diagonal blue stripes in the sidebar. (Figured out what to do.  Haven’t done it yet.) I need to see if that Search box actually works. (Done.  It works, but the page returned is ugly.) And, I’d like to substitute one of my own photographs in the banner. (Figured out what to do; haven’t done it yet.) After all that, I’ll probably change the footer to say that the design is “based on” one from ricdes rather than “Coded&Designed By.” (Done.) But, if all goes well then I don’t think all that should take too awfully long.

# Blogroll

On previous incarnations of this site, I’ve never done the blogroll thing. I decided to do it this time, because WordPress makes it so easy. At the moment, I decided to limit myself to personal weblogs of people that I know in real life. There are lots of other folks I read, but these are folks that I know in the flesh.

# Live at Mackenab.com

Well, we’re live back over here at mackenab.com.  I still haven’t been able to setup a redirect from mackenab.org, but that’s coming soon enough.  If you find anything broken please let me know.  I checked the obvious things (like images from old posts), but I’m sure I could have missed something.

One more thing before I really go live: I want to change this WordPress default theme to something a little more interesting.