Now that I am focusing on graph theory and Petri Nets, the time set aside for general discrete math topics is waning. Thus far, I have covered logic, proofs, sets, functions, relations, and a few other topics. Algorithm analysis, recursion, grammars, and related topics will be covered only if/when I discover a need for that information. Therefore, probability, counting, permutations, etc. will be the last topics of my general survey, which has been very enlightening, and a lot of fun. I decided to save these topics until the end because clinical epidemiology and biostatistics have been part of my professional life for the last 30+ years and I am comfortable with probability concepts. Live and Learn…
Ok, so I am reading my discrete math book and a few pages into the probability introduction, the “Monty Hall” problem is introduced. It is named after the famous game show host of “Let’s Make a Deal” fame. The problem is simple and maddening.
You are presented with three doors: A, B, and C. Behind one door is a car; the other two doors conceal goats. Assume for this discussion that the show’s host DOES NOT know which door is hiding the car. Now, you are asked to select a door. You arbitrarily select door “A.” The host then (randomly) selects door “C” and opens it, revealing a goat. At this point, you are given a choice to either stick with door “A” or change your selection to door “B.” Here is the million dollar question: Will switching your selection to door “B” increase your chances of winning? Alternatively, is the chance of winning by keeping door “A” the same as that of moving to door “B”?
Falling back on my many years of thinking about probability, I decided that sticking with my original choice provided the same odds of winning as switching doors would yield. This is what most people conclude, and it’s wrong.
The math book states that the chance of winning by switching to door “B” is 66.67% and by sticking with door “A” only 33%. On reading this, I was quite annoyed—it’s ridiculous. There are two doors and, obviously, the odds are 50/50 for each door–right? I actually decided to contact the book’s author and lodge a polite, but firm protest (what was I thinking??). However, caution got the best of me, and I decided to find evidence supporting my result via Google before possibly embarrassing myself. Guess what? They all agreed with the book!
After spending a good two hours or so wrestling with my disbelief, and despite having seen that every source agreed with the book, I still could not accept the stated solution. Finally, I found a page that suggested visualizing the result using a million doors (I can’t find this page now, and I was too irritated to bookmark the page during my initial visit). When I tried doing it this way (but using only five doors), I was able to come up with a solution that I could live with and embrace. Oddly, the solution is clear with many doors and, yet, counterintuitive with only three. (For those who remain skeptical, here is an interactive simulation from the NY Times.)
So what do pigeons have to do with this? One resource that appeared during my Google search was the Monty Hall Problem page on Wikipedia. (Do read this article as it is full of useful mathematics and amusing background information on this problem. If you are like me, you will take great comfort, even pleasure, in learning that many mathematicians got this wrong too!)
Among the references cited by the article was this paper reporting on an experiment conducted using human subjects and pigeons, Are Birds Smarter Than Mathematicians? Pigeons (Columba livia) Perform Optimally on a Version of the Monty Hall Dilemma. The abstract states:
A series of experiments investigated whether pigeons (Columba livia), like most humans, would fail to maximize their expected winnings in a version of the MHD. Birds completed multiple trials of a standard MHD, with the three response keys in an operant chamber serving as the three doors and access to mixed grain as the prize. Across experiments, the probability of gaining reinforcement for switching and staying was manipulated, and birds adjusted their probability of switching and staying to approximate the optimal strategy. Replication of the procedure with human participants showed that humans failed to adopt optimal strategies, even with extensive training.
Yes, you are reading this correctly. The pigeons outdid the people. Pigeons!?!?
This experience has taught me three things: 1) I need to spend more time on this chapter than I thought; 2) before firing off a snarky letter to a mathematician/book author, do some research; and 3) never
PS (9/6/120) — Apparently, the title I chose for this post is quite popular (I am not as clever as I thought), even for posts/articles that have no obvious ties to the Monty Hall Problem. I found these using Google (Link1, Link2). Perhaps we are being unfair to pigeons. After all, they seem to be holding their own…