Sunday, August 7, 2011

100 Essential Things You Didn't Know You Didn't Know by John D. Barrow

Note to all Monty Hall Problem fanatics who found this page through a search engine: My interpretation of the problem begins several paragraphs down. Please do not get bored and wander elsewhere while I'm going on about virtual monkeys.

John D. Barrow, professor of mathematics at Cambridge, has presented us laypeople with 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World.

This book is, happily, not dumbed down. He doesn’t presuppose you know calculus, but some of the 100 chapters are heavy with equations. If equations cause your eyes to fog over, you can skip past them; if they really and truly bug you, the chapters are short and you can always move on to the next one, but bear in mind that equations encode objective reality and becoming comfortable with them is one way to show respect to the universe.

I will admit my eyes fog over at the sight of long equations. This may be a factor in the unhappy fact that, despite having something of a natural aptitude for math, I crashed and burned in college calculus. I am not proud of this and I hope to overcome my foggy-mindedness.

Chapter 3 was a particular favorite of mine. We’ve all heard that with a million monkeys typing away at a million typewriters, eventually we’d get the works of Shakespeare. Only with the advent of modern supercomputers and the virtual monkeys contained therein could we actually put this to the test.

The Monkey Shakespeare Simulator Project eventually produced 10^35 pages of random typing. If you don’t have a good sense of how many pages 10^35 is, don’t worry; that only means you’re human. Human brains didn’t evolve to easily conceive of such numbers.

They checked the output against the collected works of Shakespeare, and the most substantive match was a 24-character string from Henry IV, Part 2: “RUMOUR. Open your ears;”

Then it quickly descends back into gibberish.

It’s not something that Babbage’s book deals with, but I couldn’t help but wonder what if they hadn’t limited themselves to Shakespeare (and in a case-sensitive fashion, at that). There must have been longer strings of legible English (or Spanish, French, pinyin Mandarin, etc.) in that 10^35 pages, particularly if you’re willing to overlook the occasional misspelling. What nuggets of wisdom are in there? I know the answer: none, because it’s just random typing. But if you can dredge a sufficiently auspicious longer string from the nonsense, think of the cult you could found around it.

But that's not what I want to write about today. I want to write about the topic of chapter 30, which Barrow titled "I Do Not Believe It!".

The chapter is about one of the most contentious and controversial word problems in mathematics. Although Barrow doesn't use the name, there's a good chance you know it as the "Monty Hall Problem". It's controversial because (apparently) most people find the correct answer to be so counter-intuitive and so obviously wrong that they can not accept it. Some of these people are highly educated in mathematics and consider themselves experts on probability, and they have carried on heated arguments online to argue against what is actually the correct answer.

But not me. Ever since I came across the Monty Hall Problem, I have never understood why the incorrect answer is meant to be the intuitive one. The right answer seems correct to me. It feels correct, all the way down to my bones.

I appear to be in a minority of one. Everybody else, including Barrow, thinks the wrong answer seems more correct.

I see two possibilities here:

a). I am mentally bizarre.

b). Something about the way I was originally introduced to the problem makes the right answer seem intuitive, and I have subconsciously been approaching the problem in that same way ever since.

I choose to believe b). Before I explain why, let me describe the infamous problem for everyone's edification. Since I slapped John D. Barrow's name at the top of this post, this will sound more like his representation of the problem than the most common representation, but it's in my own words.


A game show host has presented me with three identical closed boxes. There is a prize inside one of the boxes. The game show host knows which box it's in. The other two are empty. I randomly choose one of the boxes. Possibility that I am correct: 1/3.

Now the game show host opens up one of the two boxes I didn't pick. It's empty. Now there are only two boxes. I have a choice: I can stay with my original pick, or I can switch to the other remaining box.

Is it to my advantage to switch boxes?

Plausible Wrong Answer #1: No, it is not to your advantage. The prize could have been in any of the three boxes, so no matter which box you pick the probability is still 1/3.

Plausible Wrong Answer #2: No, it is not to your advantage. There is a 50/50 chance of the prize being in either of the two remaining boxes, so the odds are the same either way.

Plausible Wrong Answer #3: No, it is not to your advantage. Word problems are useless, simplified caricatures of reality and you won't get any benefit from working them out.

Actual Correct Answer: Yes, you should switch to the other unopened box. There is a 2/3 chance that the prize is in there.

At this point the audience explodes in an uproar, and Dr. Theodore Q. Figglebottom, Professor of Mathematics, proceeds to spend the next couple of hours ranting on the Internet about how everybody who thinks I should switch boxes is an uneducated fool who doesn't understand probability.

From my point of view, the Professor Figglebottoms of the world (and there are many of them) are so intent on proving themselves correct that they're missing two very important points:

1. The game show host knows which box has the prize.

2. The game show host was always going to open an empty box.

If the game show host had the same limited knowledge as me, and there was a 1 in 3 chance that he was going to wreck the whole game by opening the prize box himself, then the answer would be much different. Then there would indeed be a 50/50 chance my box had the prize, and a 50/50 chance the other box had the prize.

But he knew.

Had I chosen correctly at the beginning, then I'm just unlucky and making the best choice of action is going to lose me the game. But in the more likely event that I chose wrongly at the beginning, switching boxes now will guarantee my victory.

The game show host consolidated both of my "roads not taken" into one and gave me a guarantee against the bad outcome that would have awaited me at the end of one of those roads. Of course, I'll still lose if I chose right at the beginning, but that's why it's a game show and not a "free money" show.

And that seems intuitively obvious to me. I don't recall exactly under what circumstances I first came across the Monty Hall Problem, but I do know that the above interpretation has always been central to my understanding of it, so it was probably part of how it was originally presented to me. It's not that I'm weird; it's that in this particular problem I was innoculated early on from emphasizing the wrong part.

1 comment:

Anonymous said...

I've seen this problem before and have never understood why you should switch boxes. Your explanation was very clear. Thank you.