Editing 2545: Bayes' Theorem

{{comic
| number    = 2545
| date      = November 22, 2021
| title     = Bayes' Theorem
| image     = bayes_theorem.png
| titletext = <nowiki>P((B|A)|(A|B)) represents the probability that you'll mix up the order of the terms when using Bayesian notation.</nowiki>
}}

==Explanation==
{{incomplete|Created by <nowiki> P(d/dx x^x | d/dx x^(1/x)) </nowiki> - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

{{w|Bayes' theorem}} describes the probability of an event, given knowledge of conditions related to the event. It is typically used to update the probability that a starting condition occurred, given an outcome. This can reveal unintuitive results when the probability involved is very small. For example, when testing a large number of people for a rare disease, even a fairly accurate test will produce more false positives than the number of people actually afflicted with the disease, and hence a positive result is more likely to be false than true.

In this comic, a teacher is presenting a problem which the students are supposed to use Bayes' theorem to solve. However, the off-panel student knows that they are studying Bayes' theorem, so they use that prior knowledge to guess the usual answer to such problems. The punch line is the caption - if you know Bayes' theorem well enough, you don't need to actually calculate the probabilities.

The title text refers to the mathematical definition of Bayes' theorem: P(A | B) = P(B|A) * P(A) / P(B). Here, P(A|B) represents the probability of some event A occurring, given that B has occurred. This is often referred to as "the probability of A given B". It can be hard to remember if P(A|B) means probability of A given B, or if it's B given A, especially when talking about the probability of an earlier cause given a later effect. Randall's joke is based on this difficulty. Here P((B|A)|(A|B)) is the probability that you ''write'' (B|A) given that the correct expression is (A|B), which makes it the probability that you got the order of the notation mixed up.

==Trivia==
When this comic came out, the title text was only "P((B", the comic itself linked to [https://xkcd.com/2545/A) A)] or [https://xkcd.com/A) A)] (depending on where the comic was viewed from), and the "Black Lives Matter" image in the header was replaced by "(A", but this was quickly corrected. ([https://web.archive.org/web/20211122212442/https://xkcd.com/2545/ archive])

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[Miss Lenhart using a pointer and pointing to a white-board with statistical formulae]
:Miss Lenhart: Given these prevalences, is it likely that the test result is a false positive?
:(off-panel voice): Well, this chapter is on Bayes' Theorem, so yes.

:[Caption below the panel]:
:Sometimes, if you understand Bayes' Theorem well enough, you don't need it.

{{comic discussion}}

[[Category:Comics featuring Miss Lenhart]]
[[Category:Statistics]]