explain xkcd - User contributions [en]

1132: Frequentists vs. Bayesians

2023-03-16T22:59:35Z

Nbrader: 23:43, 12 March 2021‎ 172.69.63.27 mistakenly made exponents negative. The result of this exponentiation isn't a probabilitym it's a number of hours.

{{comic
| number = 1132
| date = November 9, 2012
| title = Frequentists vs. Bayesians
| image = frequentists_vs_bayesians.png
| titletext = 'Detector! What would the Bayesian statistician say if I asked him whether the--' [roll] 'I AM A NEUTRINO DETECTOR, NOT A LABYRINTH GUARD. SERIOUSLY, DID YOUR BRAIN FALL OUT?' [roll] '... yes.'}}

==Explanation==
This comic is a joke about jumping to conclusions based on a simplistic understanding of probability. The "{{w|base rate fallacy}}" is a mistake where an unlikely explanation is dismissed, even though the alternative is even less likely. In the comic, a device tests for the (highly unlikely) event that the sun has exploded. A degree of random error is introduced, by rolling two {{w|dice}} and lying if the result is double sixes. Double sixes are unlikely (1 in 36, or about 3% likely), so the statistician on the left dismisses it. The statistician on the right has (we assume) correctly reasoned that the sun exploding is ''far more'' unlikely, and so is willing to stake money on his interpretation.

The labels given to the two statisticians, in their panels and in the comic's title, are not particularly fair or accurate, a fact which [[Randall]] has acknowledged:<ref name="munroe-on-gelman">[http://web.archive.org/web/20130117080920/http://andrewgelman.com/2012/11/16808/#comment-109366 Comment by Randall Munroe] to "I don’t like this cartoon", blog post by Andrew Gelman in ''Statistical Modeling, Causal Inference, and Social Science''. Archived Jan 17 2013 by the Wayback Machine.</ref>
<blockquote>I seem to have stepped on a hornet’s nest, though, by adding “Frequentist” and “Bayesian” titles to the panels. This came as a surprise to me, in part because I actually added them as an afterthought, along with the final punchline. … The truth is, I genuinely didn’t realize Frequentists and Bayesians were actual camps of people—all of whom are now emailing me. I thought they were loosely-applied labels—perhaps just labels appropriated by the books I had happened to read recently—for the standard textbook approach we learned in science class versus an approach which more carefully incorporates the ideas of prior probabilities.</blockquote>

The "{{w|Frequentist inference|frequentist}}" statistician is (mis)applying the common standard of "{{w|P-value|p}}<0.05". In a scientific study, a result is presumed to provide strong evidence if, given that the {{w|null hypothesis}}, a default position that the observations are unrelated (in this case, that the sun has ''not'' gone nova), there would be less than a 5% chance of observing a result as extreme. (The null hypothesis was also referenced in [[892: Null Hypothesis]].)

Since the likelihood of rolling double sixes is below this 5% threshold, the "frequentist" decides (by this rule of thumb) to accept the detector's output as correct. The "{{w|Bayesian statistics|Bayesian}}" statistician has, instead, applied at least a small measure of probabilistic reasoning ({{w|Bayesian inference}}) to determine that the unlikeliness of the detector lying is greatly outweighed by the unlikeliness of the sun exploding. Therefore, he concludes that the sun has ''not'' exploded and the detector is lying.

A real statistician (frequentist or Bayesian) would probably demand a lower ''p''-value before concluding that a test shows the Sun has exploded; physicists tend to use 5 sigma, or about 1 in 3.5 million, as the standard before declaring major results, like discovering new particles. This would be equivalent to rolling between eight and nine dice and getting all sixes, although this is still not "very good" compared to the actual expected likelihood of the Sun spontaneously going nova, as discussed below.

The line, "Bet you $50 it hasn't", is a reference to the approach of a leading Bayesian scholar, {{w|Bruno de Finetti}}, who made extensive use of bets in his examples and thought experiments. See {{w|Coherence (philosophical gambling strategy)}} for more information on his work. In this case, however, the bet is also a joke because we would all be dead if the sun exploded. If the Bayesian wins the bet, he gets money, and if he loses, they'll both be dead before money can be paid. This underlines the absurdity of the premise and emphasizes the need to consider context when examining probability.

It is also possible that the use of the sun is a reference to Laplace's {{w|Sunrise problem}}.

The title text refers to a classic series of logic puzzles known as {{w|Knights and Knaves#Fork in the road|Knights and Knaves}}, where there are two guards in front of two exit doors, one of which is real and the other leads to death. One guard is a liar and the other tells the truth. The visitor doesn't know which is which, and is allowed to ask one question to one guard. The solution is to ask either guard what the other one would say is the real exit, then choose the opposite. Two such guards were featured in the 1986 Jim Henson movie ''{{w|Labyrinth (1986 film)|Labyrinth}}'', hence the mention of "A LABYRINTH GUARD" here. A labyrinth was also mentioned in [[246: Labyrinth Puzzle]].

===Further a less serious mathematical exploration===
As mentioned, this is an instance of the {{w|base rate fallacy}}. If we treat the "truth or lie" setup as simply modelling an inaccurate test, then it is also specifically an illustration of the {{w|false positive paradox}}: A test that is rarely wrong, but which tests for an event that is even rarer, will be more often wrong than right when it says that the event has occurred.

The test, in this case, is a neutrino detector. It relies on the fact that neutrinos can pass through the earth, so a neutrino detector would detect neutrinos from the sun at all times, day and night. The detector is stated to give false results ("lie") 1/36th of the time.

There is no record of any star ever spontaneously exploding—they always show signs of deterioration long before their explosion—so the probability is near zero. For the sake of a number, though, consider that the sun's estimated lifespan is 10 billion years. Let's say the test is run every hour, twelve hours a day (at night time). This gives us a probability of the Sun exploding at one in 4.38×1013. Assuming this detector is otherwise reliable, when the detector reports a solar explosion, there are two possibilities:
# The sun '''has''' exploded (one in 4.38×1013) and the detector '''is''' telling the truth (35 in 36). This event has a total probability of about 1/(4.38×1013) × 35/36 or about one in 4.50×1013
# The sun '''hasn't''' exploded (4.38×1013 − 1 in 4.38×1013) and the detector '''is not''' telling the truth (1 in 36). This event has a total probability of about (4.38×1013 − 1) / 4.38×1013 × 1/36 or about one in 36.

Clearly the sun exploding is not the most likely option. Indeed, Bayes' theorem can be used to find the probability that the Sun has exploded, given a result of "yes" and the prior probability given above:

: <math>
\begin{align}
P(\text{exploded}\,|\,\text{yes})&=\frac{P(\text{yes}\,|\,\text{exploded})P(\text{exploded})}{P(\text{yes})}\\
&=\frac{P(\text{exploded})(1-P(\text{lie}))}{P(\text{exploded})(1-P(\text{lie}))+P(\text{lie})(1-P(\text{exploded}))}\\
&\approx\frac1{1.25226\times10^{12}}
\end{align}
</math>

==Transcript==
:[Caption above the first panel:]
:<big>Did the sun just explode?</big>
:(It's night, so we're not sure)

:[Two Cueball-like guys stand on either side of a small table with a small black device on it. The device has white lines (ventilation) and two small antennas and a button on top. When the device speaks it uses in Westminster typeface. The Guy on the left, called Frequentist Statistician in the 2nd panel, points to the device. The guy on the right, called Bayesian Statistician in the 3rd panel, is just looking at the device. Above the spoken word from the device is a sound.]
:Frequentist Statistician: This neutrino detector measures whether the sun has gone nova.
:Bayesian Statistician: Then, it rolls two dice. If they both come up as six, it lies to us. Otherwise, it tells the truth.
:Frequentist Statistician: Let's try. ''Detector! Has the sun gone nova?''
:Sound:''Roll''
:Device: <big>YES.</big>

:[Two panels side by side are beneath the first panel. together they are broader than the top panel. Above each panel is a caption. In the left panel only the left statistician is shown with the device on the table. And in the right panel only the right statistician is shown with the device on the table. both are just looking at the device.]
:Frequentist Statistician:
:Frequentist Statistician: The probability of this result happening by chance is 1/36=0.027. Since p<0.05, I conclude that the sun has exploded.

:Bayesian Statistician:
:Bayesian Statistician: Bet you $50 it hasn't.

==Trivia==
* The Sun will never explode as a supernova, because it does not have enough mass to undergo core collapse and also does not have a companion star
*In the same blog comment as cited above<ref name="munroe-on-gelman"/>, Randall explains that he chose the "sun exploding" scenario as a more clearly absurd example than those usually used:
<blockquote>…I realized that in the common examples used to illustrate this sort of error, like the cancer screening/drug test false positive ones, the correct result is surprising or unintuitive. So I came up with the sun-explosion example, to illustrate a case where naïve application of that significance test can give a result that’s obviously nonsense.</blockquote>
*"Bayesian" statistics is named for Thomas Bayes, who studied conditional probability — the likelihood that one event is true when given information about some other related event. From {{w|Bayes Theorem|Wikipedia}}: "Bayesian interpretation expresses how a subjective degree of belief should rationally change to account for evidence".
* The "frequentist" says that 1/36 = 0.027. It's actually 0.02777…, which should round to 0.028.
* Using neutrino detectors to get an advance warning of a supernova is possible, and the {{w|Supernova Early Warning System}} does just this. The neutrinos arrive ahead of the photons, because they can escape from the core of the star before the supernova explosion reaches the mantle.

==References==
<references/>

{{comic discussion}}

[[Category:Comics featuring Cueball]]
[[Category:Multiple Cueballs]]
[[Category:Statistics]]
[[Category:Physics]]
[[Category:Astronomy]]

Talk:2663: Tetherball Configurations

2022-08-25T12:20:40Z

Nbrader: added comment on rating as well as missed opportunities

Is anyone else reminded of the "classes of a lever" sort of classification? Where the load, fulcrum, and force are permuted. I know that's not explicitly connected to this comic, but it feels like a similar vibe, since you've got 4 (or 3 out of the 4) elements, and you're just changing the order they're oriented relative to each other. Also, tempted to delete the above comment because it's neither relevant nor signed. [[User:Dextrous Fred|Dextrous Fred]] ([[User talk:Dextrous Fred|talk]]) 03:52, 25 August 2022 (UTC)

Ground-rope-ball is arguably a playable cooperative configuration. Player 1 whirls the ball above her head like a bola; Player 2 attempts to hit the ball and get it to reverse direction. Play continues until the ball hits the ground. The final score is equal to the number of reversals. [[Special:Contributions/172.70.93.43|172.70.93.43]] 06:29, 25 August 2022 (UTC)
: Ground-rope-ball is actually quite legit - I have one of these somewhere in the basement... https://www.youtube.com/watch?v=2FT0Z95kN4w [[User:Elektrizikekswerk|Elektrizikekswerk]] ([[User talk:Elektrizikekswerk|talk]]) 06:59, 25 August 2022 (UTC)
:: How does that base stay on the ground? --[[User:NeatNit|NeatNit]] ([[User talk:NeatNit|talk]]) 07:52, 25 August 2022 (UTC)
::: It's quite heavy. You could have the same result by somehow connecting the rope directly to the ground. [[User:Elektrizikekswerk|Elektrizikekswerk]] ([[User talk:Elektrizikekswerk|talk]]) 08:35, 25 August 2022 (UTC)
: Ground-rope-ball (GRB) definitely looks good. If you just place it in a playground and let some kids mess around, I guarantee they will eventually come up with rules that make for a fun game. It might not be Tetherball, but it's gotta be worthy of at least 4 stars. --[[User:NeatNit|NeatNit]] ([[User talk:NeatNit|talk]]) 07:52, 25 August 2022 (UTC)
::Not agreeing that it would work in any way related to Tetherball. But a call stuck in the ground like this would definitely get kicked by kids. So as a game it might be used, gut not as Tetherball. --[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 08:27, 25 August 2022 (UTC)
:::Though I take your point that the original comic probably intends the meaning of the rating as being "how good AS tetherball" I disagree that it's that bad at being tether ball. There is still a ball, it is tethered and you can even kick it and have it orbit back towards you. [[User:Nbrader|Nbrader]] ([[User talk:Nbrader|talk]]) 12:20, 25 August 2022 (UTC)

I feel like this comic missed some opportunities:
*Pole-Rope-Pole: Nunchuks
*Ground-Pole-Rope-Pole-Ground: Tightrope
*Pole: This configuration could be used at the same time as the above for added stability
I'm sure there are more![[User:Nbrader|Nbrader]] ([[User talk:Nbrader|talk]]) 12:20, 25 August 2022 (UTC)

In Denmark I never played this game, but often played {{w|Totem tennis}} (tether tennis or swingball). Had to find out what it was called in English first before I could write it here. --[[User:Kynde|Kynde]] ([[User talk:Kynde|talk]]) 08:27, 25 August 2022 (UTC)
:I always assumed that tetherball/swingball was effectively the same whether entirely freely pivoting/rolling-over or as the helical-track system (which just automated the 'scoring' system, and undeniably triggered the top to pop up when either limit of travel was reached) that I recall from my teen years. Not sure if it was branded to Mookie Toys, but was definitely more than a decade before the 1993 date that this article appears to suggest the helix-version was created (by some interpretations*) so it could have been amongst the properties it says they bought at that time.
:(* - I'd check exactly what it should mean and rewrite that article accordingly, but my mobile IP at any given moment is almost always on Wikipedia's no-editting list, so I'd need to wait to be tethered to a landline broadband again, and by then I'll have forgotten...)
:I also recall a 'ground weight'-tethered version (with optional peg-holes for further immobilisation if placed upon peggable ground, like your average lawn) in the box of sports equipment taken on cub-/scout-camps, which was full of many other (and often not very Health-And-Safety-compatible) outdoor 'toys' and sports equipment like lawn-darts and several rather antique-looking boxing gloves. Can't recall any branding. [[Special:Contributions/172.70.91.78|172.70.91.78]] 09:03, 25 August 2022 (UTC)
:In wikipedia it mentions something I think which is similar: "An early variant described in Jessie H. Bancroft's 1909 book Games for the Playground... involves a tethered tennis ball hit by racquets, with similar rules of the game." It sounds like this would be a rather dangerous version, with kids swinging racquets wildly in close quarters. Are there a lot of racquet-related injuries? [[User:Gbisaga|Gbisaga]] ([[User talk:Gbisaga|talk]]) 11:42, 25 August 2022 (UTC)

In France, we have "Jokari" which is pretty similar to the first scenario, except that the rope is a rubber band, played by two people. It's a bit like tennis but without the net and with a ball that comes back. Totally playable. The article on English Wikipedia is not the same thing. [[Special:Contributions/172.71.130.29|172.71.130.29]] 10:17, 25 August 2022 (UTC)

2594: Consensus Time

2022-03-16T19:05:47Z

Nbrader: expanded on consequences of moving midnight

{{comic
| number = 2594
| date = March 16, 2022
| title = Consensus Time
| image = consensus_time.png
| titletext = Now, you may argue that the varying hour lengths and feedback effects would cause chaos. To which I say, yeah, and I'm also curious to see how the weekday cycle interacts with it! So, you in?
}}

==Explanation==
{{w|Daylight Saving Time}}, which recently occurred, is frequently complained about due to having been invented for a no-longer-relevant cause. one of many complaints about this is that it will still "feel" like 5am at 6, or whatever other case. randall, partly to mock this, proposes a system that allows everybody to say when it "feels" like 9am, and then the average 9am will become the real 9am. this happens every day. as the title text points out, this would be chaotic and, to put it bluntly, awful.

Presumably the times indicated on this diagram are as the clocks in this timezone would indicate, as opposed to an "ordinary" reference time.

Although the hours between midnight and 9 am are labeled as "longer" (which we can assume means each would take longer than an hour of ordinary time to pass) the effect on the remaining hours is left unstated. If we assume that the remaining hours pass at the usual rate then this would suggest that midnight would come sooner or later than normal and hence the next vote would occur sooner or later respectively. This implies the time in this timezone could drift further than a day (or even multiple days) from existing time-zones which could be what is meant by "feedback", "chaos" and the effect on weekdays mentioned in the title text.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:Proposal: Consensus Time

:Every day, anyone in the time zone can press a button when they feel like it's 9 AM. The next day, clocks slow down or speed up to match the median choice from the previous day.

:Midnight
:6AM
:9AM today
:Median
:Noon
:6PM
:Midnight
:Longer hours
:6AM
:9AM tomorrow
:Noon
:6PM
:Midnight

:Beep
:Beep

{{comic discussion}}

2545: Bayes' Theorem

2021-11-23T13:16:01Z

Nbrader:

{{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|P(d/dx|x^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.
{| class="wikitable"
|| || Tested positive || Tested negative || Total
|-
|| Affected || 0.1% || 0.0% || 0.1%
|-
|| Unaffected || 0.9% || 99% || 99.9%
|-
|| Total || 1% || 99% || 100%
|}
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 : 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%

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 - The student doesn't need to do the calculation because they're familiar with questions involving Bayes' theorem and how they often present the counter intuitive result to illustrate the importance of prevalence to the calculation.

There is perhaps also a self-referential situation here where the student has updated their prior probabilities a number of times for whether the answer was "Yes" to a question involving Bayes' Theorem. If their method of answering "Yes" to every such question has succeeded every time before then by Bayes' theorem they will have a lot of justification to continue to do until they start getting it wrong. The prevalence of Bayes Theorem questions that require the answer "No" might be small enough that this doesn't happen in any small number of times and so they learn nothing of the false-positive rate until that point in time. This could be interpreted as a criticism of Bayesian Statistics which may treat a judgement as well justified (e.g. getting the question right) despite lacking a clear understanding of mechanism (e.g. basing your answer to the question on the numbers provided).

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==
:[Miss Lenhart is pointing with a pointer, held in her right hand, to a white-board with tables, what looks like formulae and lots of other unreadable text. She looks toward her off-panel class, from where a voice replies to her question.]
: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.

==Trivia==
*When this comic came out, the title text was only "P((B", and 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.
**See this [https://web.archive.org/web/20211122212442/https://xkcd.com/2545/ archived] version.
*It turns out that it is the notation that messes with the home page as it also messes with this wiki.
**In this [https://www.explainxkcd.com/wiki/index.php?title=2545:_Bayes%27_Theorem&oldid=221182 version] of this page, the [https://www.explainxkcd.com/wiki/index.php?title=2545%3A_Bayes%27_Theorem&type=revision&diff=221182&oldid=221181 correct title text] has been entered, but it still looked the same so everything from behind the first "|" fails to show.
**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]].
***This is extra funny since [[Blondie]], is both sometimes used for Mrs Roberts and for Miss Lenhart from this comic.

{{comic discussion}}

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

2545: Bayes' Theorem

2021-11-23T13:11:18Z

Nbrader: added meta point

{{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|P(d/dx|x^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.
{| class="wikitable"
|| || Tested positive || Tested negative || Total
|-
|| Affected || 0.1% || 0.0% || 0.1%
|-
|| Unaffected || 0.9% || 99% || 99.9%
|-
|| Total || 1% || 99% || 100%
|}
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 : 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%

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 - The student doesn't need to do the calculation because they're familiar with questions involving Bayes' theorem and how they often present the counter intuitive result to illustrate the importance of prevalence to the calculation.

There is perhaps also a self-referential situation here where the student has updated their prior probabilities a number of times for whether the answer was "Yes" to a question involving Bayes' Theorem. If their method of answering "Yes" to every such question has succeeded every time before then by Bayes' theorem they will have a lot of justification to continue to do until they start getting it wrong. The prevalence of Bayes Theorem questions that require the answer "No" might be small enough that this doesn't happen in any small number of times and so they learn nothing of the false-positive rate until that point in time.

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==
:[Miss Lenhart is pointing with a pointer, held in her right hand, to a white-board with tables, what looks like formulae and lots of other unreadable text. She looks toward her off-panel class, from where a voice replies to her question.]
: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.

==Trivia==
*When this comic came out, the title text was only "P((B", and 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.
**See this [https://web.archive.org/web/20211122212442/https://xkcd.com/2545/ archived] version.
*It turns out that it is the notation that messes with the home page as it also messes with this wiki.
**In this [https://www.explainxkcd.com/wiki/index.php?title=2545:_Bayes%27_Theorem&oldid=221182 version] of this page, the [https://www.explainxkcd.com/wiki/index.php?title=2545%3A_Bayes%27_Theorem&type=revision&diff=221182&oldid=221181 correct title text] has been entered, but it still looked the same so everything from behind the first "|" fails to show.
**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]].
***This is extra funny since [[Blondie]], is both sometimes used for Mrs Roberts and for Miss Lenhart from this comic.

{{comic discussion}}

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

2545: Bayes' Theorem

2021-11-23T13:02:31Z

Nbrader: explained joke better

{{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|P(d/dx|x^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.
{| class="wikitable"
|| || Tested positive || Tested negative || Total
|-
|| Affected || 0.1% || 0.0% || 0.1%
|-
|| Unaffected || 0.9% || 99% || 99.9%
|-
|| Total || 1% || 99% || 100%
|}
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 : 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%

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 - The student doesn't need to do the calculation because they're familiar with questions involving Bayes' theorem and how they often present the counter intuitive result to illustrate the importance of prevalence to the calculation.

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==
:[Miss Lenhart is pointing with a pointer, held in her right hand, to a white-board with tables, what looks like formulae and lots of other unreadable text. She looks toward her off-panel class, from where a voice replies to her question.]
: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.

==Trivia==
*When this comic came out, the title text was only "P((B", and 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.
**See this [https://web.archive.org/web/20211122212442/https://xkcd.com/2545/ archived] version.
*It turns out that it is the notation that messes with the home page as it also messes with this wiki.
**In this [https://www.explainxkcd.com/wiki/index.php?title=2545:_Bayes%27_Theorem&oldid=221182 version] of this page, the [https://www.explainxkcd.com/wiki/index.php?title=2545%3A_Bayes%27_Theorem&type=revision&diff=221182&oldid=221181 correct title text] has been entered, but it still looked the same so everything from behind the first "|" fails to show.
**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]].
***This is extra funny since [[Blondie]], is both sometimes used for Mrs Roberts and for Miss Lenhart from this comic.

{{comic discussion}}

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

2450: Post Vaccine Social Scheduling

2021-04-15T11:59:41Z

Nbrader:

{{comic
| number = 2450
| date = April 14, 2021
| title = Post Vaccine Social Scheduling
| image = post_vaccine_social_scheduling.png
| titletext = As if these problems weren't NP-hard enough.
}}

==Explanation==
{{incomplete|Created by a UNVACCINATED MOVIEGOER. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}
The comic shows a timeline of a multitude of (presumably) friends and acquaintances getting their two doses of vaccine. Due to the CDC-recommended delay between shots, as well as few weeks needed to build antibodies from the second shot, planning get-togethers in advance becomes complicated by who is free to meet, or not yet.

Eventually, everyone can start getting together, but during the time where some people have only received one dose, or neither dose, or their second dose recently, the scheduling is complicated.

The title text references NP-hardness, a theme that has come up in past comics. [https://en.wikipedia.org/wiki/NP-hardness NP-hardness] describes a particular level of computational difficulty. Scheduling problems are normally NP-hard. But when extra challenges such as having to deal with whether or not people are vaccinated they become even more difficult.

There appears to be an error: The third person is shown as having received their second dose twice.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
A time graph of eleven people with lines. Circles with 1 and 2 are interspaced on the lines to represent first and second doses for COVID, and bolded lines for times after their second doses. 6 ellipses intersect various subsets of the people labeled in order:

DINNER GAMES MOVIE BIRTHDAY DINNER CABIN

[Caption below the panel]:

Post Vaccine Social Scheduling
{{comic discussion}}
[[Category: COVID-19]]
[[Category: COVID-19 vaccine]]
[[Category:Timelines]]