Bayesed and Confused?

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.)


  1. 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.

    Posted May 8, 2016 at 5:33 pm | Permalink
  2. Harold says

    Monte Hall: Your first choice is probably wrong; if it is and you switch you win.

    Posted May 9, 2016 at 3:13 am | Permalink
  3. JK says

    Uhm Henry, that Priori thingy – that a Prius without the battery?

    (My thinking being along the lines of radius/radii.)

    Posted May 9, 2016 at 10:21 am | Permalink
  4. Silly me, thinking this was a serious thread.

    Bayesian, schmayesian. Who really gives a shit, am I right?

    Posted May 9, 2016 at 12:17 pm | Permalink
  5. Malcolm says

    Henry – that was an excellent breakdown you gave of the Monty Hall problem, really the clearest and most concise I’ve seen.

    Posted May 10, 2016 at 10:18 am | Permalink
  6. Thank you, Malcolm.

    Posted May 10, 2016 at 12:55 pm | Permalink
  7. 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:

    Bayes’s main idea, known simply as Bayes’s Theorem, is a way to think about credences [the term professional statisticians use for degrees of belief]. It allows us to answer the following question. Imagine that we have certain credences assigned to different beliefs. Then we gather some information, and learn something new. How does that new information change the credences we have assigned? That’s the question we need to be asking ourselves over and over, as we learn new things about the world.

    Posted May 12, 2016 at 8:26 pm | Permalink

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