Editing 2545: Bayes' Theorem
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==Explanation== | ==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.}} | ||
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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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|| Total || 1% || 99% || 100% | || Total || 1% || 99% || 100% | ||
|} | |} | ||
− | For example, if a test has a 100% sensitivity ( | + | For example, if a test has a 100% sensitivity (all infected are tested positive) and a 99% specificity (1% of unaffected nevertheless are 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) the subject is unaffected nine time out of ten. Although this would be a very performant test, given the prevalence, chances are in fact that the test is a false positive. |
− | For this same example, the Bayesian formula gives | + | For this same example, the Bayesian formula gives : P( Affected | Positive ) = P( Positive | Affected ) * P( Affected ) / P( Positive ) = 100% * 0.1% / 1% = 10% and P( Unaffected | Positive ) = P( Positive | Unaffected ) * P( Unaffected ) / P( Positive ) = 0.9009% * 99.9% / 1% = 90% |
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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 understand Bayes' theorem well enough, you know where your intuition is wrong and 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. |
==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]: | ||
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**Now this has [https://www.explainxkcd.com/wiki/index.php?title=2545%3A_Bayes%27_Theorem&type=revision&diff=221183&oldid=221182 been fixed] using the <nowiki><nowiki></nowiki> format. | **Now this has [https://www.explainxkcd.com/wiki/index.php?title=2545%3A_Bayes%27_Theorem&type=revision&diff=221183&oldid=221182 been fixed] using the <nowiki><nowiki></nowiki> format. | ||
***Seems like [[Randall]] made an exploit on himself like [[Mrs. Roberts]] did in [[327: Exploits of a Mom]]. | ***Seems like [[Randall]] made an exploit on himself like [[Mrs. Roberts]] did in [[327: Exploits of a Mom]]. | ||
− | ***This is extra funny since [[Blondie]] is both sometimes used for Mrs Roberts and for Miss Lenhart from this comic. | + | ***This is extra funny since [[Blondie]], is both sometimes used for Mrs Roberts and for Miss Lenhart from this comic. |
{{comic discussion}} | {{comic discussion}} |