24 August 2008

The Monty Hall Puzzle, continued

Most people, when confronted with the story in yesterday's entry, will answer that there is no reason to change, that the contestant might as well stick with curtain A. After all, there are two unopened curtains. The big prize is behind one of them. The odds that it is behind curtain number one are, then, 0.50, or 1:2, so the rational contestant is indifferent as between one curtain and the other.

Conditional probability theory, though, looks carefully at what Monty Hall's own actions have already told us. When the contestant made her first choice her odds of having picked the right curtain were: one out of three. The chance that she had made the wrong choice then, was: two out of three. What Mr. Hall has told us hasn't changed the chance that her initial choice was wrong at all—it has concentrated that chance—which is now embodied, so to speak, by curtain B. Why? Because Mr. Hall knows where the $1 million is, and surely wouldn't have opened curtain three if he knew it was there. So his choice to open the one that he did was non-random.

At any rate, conditional probability theory says that given the fact that Monty now has eliminated curtain C the rational contestant will pick curtain B, giving herself a two-thirds chance of winning. Don't be surprised if this is counter-intuitive and even disorienting. It has that effect on a lot of people. As a psychological matter, if she picked curtain A, and Monty opened curtain C, she might well take that as a confirming event ("so far, what has happened is consistent with my initial guess") which would make her likely to dig in her heels and stick with it, theory be damned!

I mention it because conditional probability theory has important consequences in the world of finance and may have something to tell us about last year's credit crunch. But I'll give the connection some thought before pontificating further,

No comments:

Knowledge is warranted belief -- it is the body of belief that we build up because, while living in this world, we've developed good reasons for believing it. What we know, then, is what works -- and it is, necessarily, what has worked for us, each of us individually, as a first approximation. For my other blog, on the struggles for control in the corporate suites, see www.proxypartisans.blogspot.com.