Bayes and the challenge about Morpheus
More than a year ago, on April 17, 2014, this article appeared on Habré (to the day of the death of Thomas Bayes). There was a lot of interesting things there, but, as usual, most of the discussions in the comments came down to a puzzle that was just an epigraph. Then I looked through the article diagonally, and ignored the task at all.
And in vain ...
Just now, having decided to understand the Bayes theorem, I began to read an intuitive explanation of an intuitive explanation . Predictably, no one has finished reading anything, but with great pleasure he undertook to solve problems. The problem of sick Sisi from the above tutorial turned out to be relatively simple without any theorems, and since my answer coincided with the author, deciding to amuse my ego, I went for additional tasks. The first problem that came across from SUBJ was about Morpheus. She was a little more cunning than the previous one, but in the end I got the answer. SUDDENLY my answer did not coincide with any of the answers given by habrovchanami in the comments to the article about Bayes. The rest of the Internet, although they gave a certain number of references to this task, did not contain anything adequate in the solutions.
To check the answer, I put a simple python script that confirmed the correctness of my decision (Percent to 99. All the same, a pseudo-random generator leads to a slight variation in the results).
')
So, on the Internet, someone is wrong, but Icannot comment on Habré (I can already, thanks). So here's your “solution”, comments, conclusions and script in the sandbox.
Once again the task (I remind the source . ):
For starters, the script .
This publication (and it would be nice to comment) I want to finally close the topic of this puzzle on the Internet.
Maybe someone will find errors in my decision or reasoning?
EDIT2:
After reading the comments my opinion about the task has changed. Now from the mathematical, the task has become a philological. Or even more theological.
Basically, all were divided into two groups:
1) Egg must be broken from the blunt side
"You close your eyes and take a pill" means:
1) The chance of getting any of the previously mentioned pills is equally likely (solution above)
2) Since the hands are previously mentioned, the algorithm for selecting tablets:
- hand selection
- Pill selection from a specific hand
And now my personal IMHO:
From the point of view of the Russian language, as well as with the goal of minimally distorting the problem with assumptions when solving - our first option is everything. But given the context (Bayes), the author probably still wanted to take into account the choice of hand. Then the problem will be dependent events and we need a theorem to solve.
And in vain ...
Just now, having decided to understand the Bayes theorem, I began to read an intuitive explanation of an intuitive explanation . Predictably, no one has finished reading anything, but with great pleasure he undertook to solve problems. The problem of sick Sisi from the above tutorial turned out to be relatively simple without any theorems, and since my answer coincided with the author, deciding to amuse my ego, I went for additional tasks. The first problem that came across from SUBJ was about Morpheus. She was a little more cunning than the previous one, but in the end I got the answer. SUDDENLY my answer did not coincide with any of the answers given by habrovchanami in the comments to the article about Bayes. The rest of the Internet, although they gave a certain number of references to this task, did not contain anything adequate in the solutions.
To check the answer, I put a simple python script that confirmed the correctness of my decision (Percent to 99. All the same, a pseudo-random generator leads to a slight variation in the results).
')
So, on the Internet, someone is wrong, but I
Once again the task (I remind the source . ):
Morpheus’s left hand contains 7 blue and 3 red tablets, and 5 blue and 8 red tablets in his right hand. You close your eyes and take a pill - it turns out to be red, but you don’t know from which hand you took it. What is the probability that you took it from the right hand?
For starters, the script .
An important point regarding conditional terms
During the discussion of the problem, there is often a tendency to take for granted that the probability of obtaining a 50/50 tablet from the right and left hand. In fact, in the task itself this is not said anywhere. We intuitively expect such an approach to the method of choosing a tablet from the hand. In practice, however, nothing was said about how the tablet was chosen. So you shouldn't invent a gag.
EDIT:
I have indicated in the comments below that this is still not convincing, so I will try to add this explanation (from the same comment):
Suppose we abstract at the hands. Let the task sound like this: On the table are 8 whole apples, 3 bitten apples, 5 whole peaches and 7 bitten peaches.
You close your eyes and take the fruit. This is an Apple. What is the probability that it is an integer?
In this problem, the question of the method of choice does not arise? Any fruit has an intuitively equal chance of being picked.
In fact, the task has not changed. Integrity is an attribute corresponding to the right hand. Biting - left. The wording with the hands makes the search for the selection method tied to the number of hands, but it seems to me - this is a mistake.
EDIT:
I have indicated in the comments below that this is still not convincing, so I will try to add this explanation (from the same comment):
Suppose we abstract at the hands. Let the task sound like this: On the table are 8 whole apples, 3 bitten apples, 5 whole peaches and 7 bitten peaches.
You close your eyes and take the fruit. This is an Apple. What is the probability that it is an integer?
In this problem, the question of the method of choice does not arise? Any fruit has an intuitively equal chance of being picked.
In fact, the task has not changed. Integrity is an attribute corresponding to the right hand. Biting - left. The wording with the hands makes the search for the selection method tied to the number of hands, but it seems to me - this is a mistake.
Decision and conclusions
And this is how I described it on a human:
The tablet is already there. Red The probability that I take a pill will get red - 47.8% (11 out of 23). The probability that when choosing a pill I take it from my left hand is 43.48% (10 of 23). From the right - 56.52% (100% - 43.48%). The probability that the tablet is taken will be from the left hand and will be red - 13.04%. The probability that the tablet is taken will be from the right hand and will be red - 34.76%. The total probability of taking the red pill is 47.8% (11 out of 23 and 13.04% + 34.76%). The probability that the red pill was taken from the right hand was 72.72% (34.76% of 47.8%. Percentage of the probability of taking the red pill from the right of the probability of getting the red pill at all).
And so I realized that all this is heresy:
Having received the answer, I noticed that it coincides one to one with the total share of red pills in the right hand (8/11). Coincidence? I didn’t want to recount, so I just began to change the number of blue tablets in the right and left hand in the script. The answer has not changed. *** - I thought. Obviously, the fact that we already have a red pill in our hand simply discards all cases when we chose the blue one. Therefore, what they are, what they are not - is monopenisual. The funny thing is that all the calculations from my solution above are correct. And continue to remain true when changing the number of any pills.
Findings:
1) No Bayes in this problem and does not smell. There is no need to apply his theorem.
2) The author of the task is a notable troll.
3) Want to make fun of someone? Throw him this puzzle and giggle viciously, while the victim is straining to try to understand what is wrong here.
four)…
5) PROFIT
Here is the solution process for me:
3/10 30%
8/13 61.5%
11/23 47.8%
3/11 27.27%
8/11 72.73%
13/23 56.52%
10/23 43.48%
30% * 43.48% / 100 = 13.04%
61.5% * 56.52% / 100 = 34.76%
34.76% * 100 / 47.8% = 72.72%
8/13 61.5%
11/23 47.8%
3/11 27.27%
8/11 72.73%
13/23 56.52%
10/23 43.48%
30% * 43.48% / 100 = 13.04%
61.5% * 56.52% / 100 = 34.76%
34.76% * 100 / 47.8% = 72.72%
And this is how I described it on a human:
The tablet is already there. Red The probability that I take a pill will get red - 47.8% (11 out of 23). The probability that when choosing a pill I take it from my left hand is 43.48% (10 of 23). From the right - 56.52% (100% - 43.48%). The probability that the tablet is taken will be from the left hand and will be red - 13.04%. The probability that the tablet is taken will be from the right hand and will be red - 34.76%. The total probability of taking the red pill is 47.8% (11 out of 23 and 13.04% + 34.76%). The probability that the red pill was taken from the right hand was 72.72% (34.76% of 47.8%. Percentage of the probability of taking the red pill from the right of the probability of getting the red pill at all).
And so I realized that all this is heresy:
Having received the answer, I noticed that it coincides one to one with the total share of red pills in the right hand (8/11). Coincidence? I didn’t want to recount, so I just began to change the number of blue tablets in the right and left hand in the script. The answer has not changed. *** - I thought. Obviously, the fact that we already have a red pill in our hand simply discards all cases when we chose the blue one. Therefore, what they are, what they are not - is monopenisual. The funny thing is that all the calculations from my solution above are correct. And continue to remain true when changing the number of any pills.
Findings:
1) No Bayes in this problem and does not smell. There is no need to apply his theorem.
2) The author of the task is a notable troll.
3) Want to make fun of someone? Throw him this puzzle and giggle viciously, while the victim is straining to try to understand what is wrong here.
four)…
5) PROFIT
This publication (and it would be nice to comment) I want to finally close the topic of this puzzle on the Internet.
Maybe someone will find errors in my decision or reasoning?
EDIT2:
After reading the comments my opinion about the task has changed. Now from the mathematical, the task has become a philological. Or even more theological.
Basically, all were divided into two groups:
"You close your eyes and take a pill" means:
1) The chance of getting any of the previously mentioned pills is equally likely (solution above)
2) Since the hands are previously mentioned, the algorithm for selecting tablets:
- hand selection
- Pill selection from a specific hand
And now my personal IMHO:
From the point of view of the Russian language, as well as with the goal of minimally distorting the problem with assumptions when solving - our first option is everything. But given the context (Bayes), the author probably still wanted to take into account the choice of hand. Then the problem will be dependent events and we need a theorem to solve.
The correct answer is the 2nd option (for descendants)
67.2%
Source: https://habr.com/ru/post/264997/
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