Friday, November 14, 2008

Bayesian Solutions to Everyday Problems

Ok. This is a new theme i have been considering writing about. It should be fun but what i need are evveryday problems for which information is uncertain and a decision must be made. Here is a very simple and oft discussed example. Suppose that you are on the show Let's Make a Deal. The scenario is as follows. You are shown three curtains and told behind the curtains are three prizes. One of the prizes is valuable the other two are generally not valuable. You are asked to pick a curtain. At this point you have a 1/3 chance of having selected the curtain with the prize behind it. You are then shown that behind one of the curtains that you did not choose is one of the crummy prizes. You are then asked whether you would like to stick with the curtain you initially chose or switch and choose the other remaining curtain.

If you stick with your original choice you still have a 1/3 chance of winning. On teh other hand, the prize had a 2/3 chance of being in one of teh other two curtains. This remains true even after you have been shown that a crummy prize was behind one of those two. As a result, the remaining curtain of the two still has a 2/3 chance of having the prize. Thus switching doubles your chances of winning. Neato!


D.J. said...

Actually, if one curtain you know for certain harbors a crummy prize, you have 2 curtains left, one of which is crummy and one of which is awesome, and there is a 50-50 chance no matter which of the remaining 2 you choose. Wouldn't it suck if you changed curtains, then realized the real prize was behind curtain number 1 afterall?

kristina said...

I would say something about Bayesian Inference, but I have heard the introductory stuff from the friend I recommended to you on facebook, and haven't given it a tremendous amount of thought since. I will note that I fully advocate your use of the terms "crummy" and "neato". I might also suggest that you refer to those who commit acts that could be described as crummy as "crumbums".

I also check my sanity with graphing calculators. Unfortunately, I rarely see misrepresented hyperbolic trig functions in my peripheral vision while having panic attacks. Damn.

SuperFlunky said...

No actually it is a 2/3 chance of winning if you switch.

However, even if you lose when you switch, it might suck that you lost, but it would suck optimally :).

Kidding aside, as with all probabilistic arguments they are only about maximizing your odds, they guarantee nothing, unless, of course, you play the game enough times.