An Intuitive (and Short) Explanation of Bayes' Theorem

Bayes’ theorem was the subject of a detailed article. The essay is good, but over 15,000 words long — here’s the condensed version for Bayesian newcomers like myself:

This is a companion discussion topic for the original entry at

you have a typo, in …

9.6% of mammograms miss breast cancer when it is there (and therefore 90.4% say it is there when it isn’t).

… you meant to say somthing like :

9.6% of mammograms incorrectly indicate breast cancer when it isn’t there, and the other 90.4% correctly say it is not there when, well, it is not there.

Thanks Gavrilo – I just fixed it.

Hey, here’s an interesting bayes problem i came across first in a book (The curious incident of the dog in the night time).

Suppose you are in a game show. You are given the choice of three doors – one of which conceals a valuable prize and the
others conceal a goat.

After you make a choice, the host opens one of the other doors (–one without a prize).

He then gives you the option of staying with the initial choice of door or switching to the other door. The door finally chosen is then opened.

Should you switch, not switch, or does it make no difference what the contestant does?

by Bayes theorum you can see that if you switch u’d have a 2:1 advantage.

Hi Amal, thanks for dropping by. Yes, I like that question too, it was presented to us as “The Monty Hall” problem when studying computer science.

It’s pretty amazing how counter-intuitive the results can be – switching your choice after you’ve picked “shouldn’t” change your chances, right? I plan on writing about this paradox, too :slight_smile:

Oddly useful! I’ve been reading Bayes explanations for a while, and this one really hit home for me for some reason.

One thing that you might consider adding (something I’ve never seen) is a pie-chart visualization of what’s going on. Basically, you have a pie of 100% of people. 1% of that pie has cancer, so that’s a tiny slice. The test will produce a positive for 80% of that 1% slice + 9.6% of the remaining 99% slice-- you can imagine that as a little blue translucent piece of appropriate size that covers most of the 1% slice and a chunk of the 99% slice. From that mental image, it’s obvious what’s going on-- there’s a lot more blue on the 99% than on the 1%. Might be too complicated, but hey. :slight_smile: Anyways, thanks.

Hey Ed, thanks for the comment. I agree – some type of chart may make the relationship that much clearer. Appreciate the suggestion, I’ll put one together.

Bayes theorem can also be thought of as

      True Positives

True Positivees + False Positives

So a large number of false positives reduces the accuracy of the test because the denominator increases.

Thanks Lee! That’s a great way to put it.

About Monty Hall- the Bayes application to this seems very forced. The Monty Hall problem is a simple probability problem, or it can be viewed as a partitioning problem. See:

Using Bayes for this makes it needlessly complex, not “betterExplained”.

Similarly, the article above is needlessly complex- nuke the first equation and leave the simpler one. You just pulled it out of thin air anyway- it doesn’t help anyone.

The usual diagram, given in HS stats classes, is a rectangle, with A, ~A on the top, B, ~B on the side. Say A is .9 and B is .2. The area of the small quadrant (.02), is the probability of A and B both happening. This area can be also viewed as P(A|B)*P(B) or P(B|A)*P(A). You have to explain why, but it’s pretty evident from the diagram. Then just equate these two and divide by P(B) and you have the simpler equation.

Hi Randy, Bayes may be overkill for the Monty Hall problem, but it’s interesting to see that it can apply there as well.

Yes, the diagram you mention may be a helpful addition to the discussion above, appreciate the feedback.

Just wanna say thank you for writing this. I know about the original article and I tried reading it but somewhere along the way I got lost and couldn’t follow it.

Hi numerodix, you’re welcome – I found the original article interesting but a bit long as well, so I decided to summarize it here.

Пример ведения IT блога…

Я редко выделяю какой-то отдельный англоязычный блог, но вот этот, BetterExplained , заслуживает внимания…

[…] An Intuitive (and Short) Explanation of Bayes’ Theorem […]

Hello, I just came upon this site and I’m finding it beautiful. I think I spotted an error in this article, though.
When you say:

“”Of those 100, 80 will test positive” rather than “80% of the 1% will test positive”).”,

you probably wanted to say: “rather than ‘80% of the 100% will test positive’”.

Hi Matteo, thanks for the comment. The statement actually refers to the original 1%, so it’s giving a way of giving compound percentages (80% of 1% vs. 80 out of 10,000).

wow thank you so much for this, you really did a good job explaining it, i have my AP statistics exam today at noon so this might save me :slight_smile:

Randy, back on Nov 7 2007, suggested using overlapping rectangles - Venn diagrams - to help clarify the Rev. Bayes. In their book “Chances Are…” (Viking Penguin, 2006), Kaplan & Kaplan did so on pp. 184 ff. Indeed it does help.

you. are. the. best.