Editing 2545: Bayes' Theorem
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==Explanation== | ==Explanation== | ||
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{{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. | {{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. | ||
{| class="wikitable" | {| class="wikitable" | ||
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|| Unaffected || 0.9% || 99% || 99.9% | || Unaffected || 0.9% || 99% || 99.9% | ||
|- | |- | ||
| − | || | + | || Test result || 1% || 99% || 100% |
|} | |} | ||
| − | For example, if a test has a 100% | + | For example, if a test has a 100% sensibility (all affected are tested positive) and a 1% rate of false positive (1% of unaffected is nevertheless tested positive), the interpretation of a positive test depends on the prevalence (percentage of affected). In the example case, the prevalence is 0.1%, so that when the test result is positive (left column) chances are in fact that the subject is unaffected nine time out of ten. Although this would be a very performant test, given the prevalence, chances are that the test is a false positive. |
| − | + | Bypassing the graphical display, the bayesian formula would give : p( Affected | Positive ) = p( Positive | Affected )*p( Affected )/p( Positive ) = 100% * 0.1% / 1% = 10% - QED. | |
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| − | 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 - | + | 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 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. |
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| + | ==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== | ==Transcript== | ||
| − | :[Miss Lenhart | + | {{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? | :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]: | :[Caption below the panel]: | ||
:Sometimes, if you understand Bayes' Theorem well enough, you don't need it. | :Sometimes, if you understand Bayes' Theorem well enough, you don't need it. | ||
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{{comic discussion}} | {{comic discussion}} | ||
