You’ve probably heard of Bayes’s Theorem, but if you’ve yet to get your head around it, here’s a nice visual explanation, including a simple Bayesian explanation of the perplexing “Monty Hall problem” (which we last discussed in here way back in 2009).
(Also, from the same website, here’s another Bayes tutorial.)
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With respect to the “Monte Hall” problem, I find it easier to explain as follows (as opposed to the visual explanation):
A priori, whichever door the contestant chooses has a 1/3 probability of being the “car-door”. Now, if Monte Hall asks you if you would swap your initial door choice for both the other two doors, you would definitely do so because the a priori chance that the car is behind one of 2 doors is twice the probability that it is behind just the 1 door of your initial choice. This is obvious despite the fact that at least 1 of those 2 doors must be a goat-door (because there is only one car-door).
In the contest, however, Monte first shows you a goat-door by opening one of the two doors you haven’t chosen. Well, we already knew that at least one of those doors had to be a goat-door (and we knew that Monte would have to show you a goat-door, not a car-door). Hence, the probability that 1 of those 2 doors not initially chosen by the contestant is indeed the car-door remains unchanged (2/3). The only change is that the entire 2/3 probability has been shifted to the door not chosen by the contestant and not opened by Monte.
Monte Hall: Your first choice is probably wrong; if it is and you switch you win.
Uhm Henry, that Priori thingy – that a Prius without the battery?
(My thinking being along the lines of radius/radii.)
Silly me, thinking this was a serious thread.
Bayesian, schmayesian. Who really gives a shit, am I right?
Henry – that was an excellent breakdown you gave of the Monty Hall problem, really the clearest and most concise I’ve seen.
Thank you, Malcolm.
I am currently reading “The Big Picture: On the Origins of Life, Meaning, and the Universe Itself” by Sean Carroll, which I recommend highly to anyone interested in such subjects. Chapter 9, titled “Learning about the World”, comprises an excellent 10-page introduction to Thomas Bayes’s Theorem, without the use of a single equation. Here is an excerpt from that chapter: