1132: Frequentists vs. Bayesians - Revision history

92.23.2.228: /* Explanation */ Correcting punctuation and capitalising tag. (Capitalising the proper names, while I'm here....)

2025-08-18T22:41:59Z

‎Explanation: Correcting punctuation and capitalising tag. (Capitalising the proper names, while I'm here....)

89.139.10.6: /* Explanation */

2025-08-18T22:13:55Z

‎Explanation

108.162.242.35: Fix unclear wording

2025-03-12T19:44:00Z

Fix unclear wording

141.101.99.90: Undo revision 313234 by 172.70.85.169 (talk) Fairly good point, but the Moon might not have risen (if not a full-moon), and editing errors don't help. Try again?

2023-05-16T13:53:29Z

Undo revision 313234 by 172.70.85.169 (talk) Fairly good point, but the Moon might not have risen (if not a full-moon), and editing errors don't help. Try again?

172.70.85.169: /* Explanation */

2023-05-16T08:51:27Z

‎Explanation

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.

2023-03-16T22:59:35Z

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.

162.158.238.7: Wrong calculation

2023-02-13T21:11:48Z

Wrong calculation

172.71.114.161: /* Explanation */ fix misinterpretation of p-value

2022-11-09T13:45:34Z

‎Explanation: fix misinterpretation of p-value

Donald Trump: Reverted edit by anti-crap user

2022-05-26T19:04:26Z

Reverted edit by anti-crap user

Donald Trump: Reverted edit by anti-crap user

2022-05-26T19:04:22Z

Reverted edit by anti-crap user

@@ Line 7: / Line 7: @@
 ==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{{citation needed}}, and so is willing to stake money on his interpretation.
+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,{{Citation needed}} 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 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]].)
+The "{{w|Frequentist inference|frequentist}}" statistician is 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.
+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.
+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}}.
+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]].
@@ Line 27: / Line 27: @@
 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.
+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 4.38×10<sup>13</sup>. Assuming this detector is otherwise reliable, when the detector reports a solar explosion, there are two possibilities:
+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 4.38×10<sup>13</sup>. Assuming this detector is otherwise reliable, when the detector reports a solar explosion, there are two possibilities:
-# The sun '''has''' exploded (one in 4.38×10<sup>13</sup>) and the detector '''is''' telling the truth (35 in 36). This event has a total probability of about 1/(4.38×10<sup>13</sup>) × 35/36 or about one in 4.50×10<sup>13</sup>
+# The Sun '''has''' exploded (one in 4.38×10<sup>13</sup>) and the detector '''is''' telling the truth (35 in 36). This event has a total probability of about 1/(4.38×10<sup>13</sup>) × 35/36 or about one in 4.50×10<sup>13</sup>
-# The sun '''hasn't''' exploded (4.38×10<sup>13</sup> − 1 in 4.38×10<sup>13</sup>) and the detector '''is not''' telling the truth (1 in 36). This event has a total probability of about (4.38×10<sup>13</sup> − 1) / 4.38×10<sup>13</sup> × 1/36 or about one in 36.
+# The Sun '''hasn't''' exploded (4.38×10<sup>13</sup> − 1 in 4.38×10<sup>13</sup>) and the detector '''is not''' telling the truth (1 in 36). This event has a total probability of about (4.38×10<sup>13</sup> − 1) / 4.38×10<sup>13</sup> × 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:
+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>

@@ Line 7: / Line 7: @@
 ==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.
+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{{citation needed}}, 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>

@@ Line 12: / Line 12: @@
 <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]].)
+The "{{w|Frequentist inference|frequentist}}" statistician is 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.

@@ Line 42: / Line 42: @@
 \end{align}
 </math>
 ==Transcript==

@@ Line 42: / Line 42: @@
 \end{align}
 </math>
 ==Transcript==

@@ Line 31: / Line 31: @@
 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 4.38×10<sup>-13</sup>. Assuming this detector is otherwise reliable, when the detector reports a solar explosion, there are two possibilities:
 # The sun '''has''' exploded (one in 4.38×10<sup>-13</sup>) and the detector '''is''' telling the truth (35 in 36). This event has a total probability of about 1/(4.38×10<sup>-13</sup>) × 35/36 or about one in 4.50×10<sup>-13</sup>
-# The sun '''hasn't''' exploded (4.38×10<sup>-13</sup> − 1 in 4.38×10<sup>-13</sup>) and the detector '''is not''' telling the truth (1 in 36). This event has a total probability of about (4.38×10<sup>-13</sup> − 1) / 4.38×10<sup>-13</sup> × 1/36 or about one in 36
+# The sun '''hasn't''' exploded (4.38×10<sup>-13</sup> − 1 in 4.38×10<sup>-13</sup>) and the detector '''is not''' telling the truth (1 in 36). This event has a total probability of about (4.38×10<sup>-13</sup> − 1) / 4.38×10<sup>-13</sup> × 1/36 or about one in 36 (this is a negative number)
 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:

@@ Line 1: / Line 1: @@
 {{comic
-| number    = 113
+| number    = 1132
-| date      = Nvember 9, 2012
+| date      = November 9, 2012
-| title     = Frequntists vs.Bayesians
+| 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 NEUTRIN DETECTOR, NOT A LABYRINTH GUARD. SERIOUSLY, DID YOUR BRAIN FALL OUT?' [roll] '... yes.'}}
+| 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 likel. In the comic, a device tests for the (highly unlikely) event that the sun has exploded. A degree of radom error is introduced, by rolling two {{w|dice}} and lying if the result is doubl sixes. Double sixes are unlikely (1 in 36, o 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.
+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/20130117080920http://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>
+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 is less than a 5% chance that the result was merely random. (The null hypothesis wa also referenced in [[892: Null Hypothesis]].)
+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 is less than a 5% chance that the result was merely random. (The null hypothesis was also referenced in [[892: Null Hypothesis]].)
-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 etween eight and nine dice and getting all sixes, although this is still not "very good" compared to the atual expected likelihood of the Sun spontaneously going nova, as discussed below.
+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.
-The line, "Bet you $50 it hasnt", 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.
+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.
 It is also possible that the use of the sun is a reference to Laplace's {{w|Sunrise problem}}.
@@ Line 28: / Line 29: @@
 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). his gives us a probability of the Sun exploding at one in 4.38×10<sup>-13</sup>. Assuming this detector is otherwise reliable, when the detector reports a solar explosion, there are two possibilities:# The sun '''has''' exploded (one in 4.38×10<sup>-13</sup>) and the detector '''is''' telling the truth (35 in 36). This event has a total probability of about 1/(4.38×10<sup>-13</sup>) × 35/36 or about one in 4.50×10<sup>-13</sup>
+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 4.38×10<sup>-13</sup>. Assuming this detector is otherwise reliable, when the detector reports a solar explosion, there are two possibilities:
-# The sun '''hasn't''' exploded (4.38×10<sup>-13/sup> − 1 in 4.38×10<sup>-13</sup>) and the detector '''is not''' telling the truth (1 in 36). This event has a total probability of about (4.38×10<sup>-13</sup> − 1) / 4.38×10<sup>-13</sup> × 1/36 or about one in 36
+# The sun '''has''' exploded (one in 4.38×10<sup>-13</sup>) and the detector '''is''' telling the truth (35 in 36). This event has a total probability of about 1/(4.38×10<sup>-13</sup>) × 35/36 or about one in 4.50×10<sup>-13</sup>
 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:
@@ Line 49: / Line 51: @@
 :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 Statstician: Let's try. ''Detector! Has the sun gone nova?''
+: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  caption. In the left panel only the left statistician is shown with the device on the table. And in the right pane only the right statistician is shown with the device on the table. both are just looking at the device.]
+:[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.
@@ Line 66: / Line 68: @@
 *"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 neutrin detectors to get an advance warning of a supernova is possible, and the {{w|Supernova Early Warning Sysem}} does just this. The neutrinos arrive ahead of the photons, because they can escape from the core of the star beore the supernova explosion reaches the mantle.
+* 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==
@@ Line 73: / Line 75: @@
 {{comic discussion}}
-[[CategoryComics featuring Cueball]]
+[[Category:Comics featuring Cueball]]
 [[Category:Multiple Cueballs]]
-[[Category:Statstics]]
+[[Category:Statistics]]
 [[Category:Physics]]
-[[Cateory:Astronomy]]
+[[Category:Astronomy]]