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In the spirit of the article we're discussing, I wrote the present comment to 1) teach something about mathematics and probability 2) share a bit of social enlightenment.
In summary this comment should change your thinking fundamentally. You will need to read it carefully but I promise it is relevant. (Please endorse it and upvote it, if you agree.)
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Program of study for this comment.
I suggest you go through this comment as follows.
1. Read section 1 (approx. 1 hour.)
Goal: improve your mathematical reasoning.
2. 30 minute break.
3. Read section 2 (approx 5 minutes).
Goal: social insight.
4. Generalize the insight just gained.
Goal: make the example more practical.
This is an exercise for the reader.
1. One of the most important mathematical videos you will ever see.
Firstly, unless you are a practicing mathematician this is one of the most important mathematical video you will ever see:
https://www.youtube.com/watch?v=BrK7X_XlGB8
Watch it. As a result you will improve your rational thinking forever.
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Break. I suggest you next take a 30-minute break. During this time you can reflect on and assimilate the knowledge you have learned.
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2. An important social insight.
This section requires you to understand section 1.
Next, suppose that you are perfectly rational. We will introduce an extreme case, and you will have to generalize it yourself, to come up with the social insight I promised.
Base (extreme) case.
If I present you a (fair) coin and ask you to judge whether it is fair, after, say, thirty or forty flips you will conclude that it is likely fair. You can never be sure, of course, but you will have a high confidence.
However, if I give you the same coin but also the knowledge that it was drawn at random from an infinite bag with 1 fair coin in it -- for example, let's say coins are numbered, I select a real number at random between 0 and 1, and only the coin with the exact value 0.5 is fair, any other coin is unfair, weighted - then even given hours, days, weeks, or years of flipping, you will come to me with the same conclusion: there is a 0% chance (you will have 0% confidence) that the coin is fair and 100% chance that the coin is weighted.[0] If I bet you a thousand to one that it is fair, you would put any amount of money that it is weighted: regardless of the amount of testing you did and the results of your tests.
This includes your running every test for randomness, flipping it millions of times and analyzing the result, anything you want.
So let's look at what happened. You have been moved from being able to quickly decide whether a coin is likely fair, to being completely unable to accept that the coin is fair. No matter how much evidence you can collect, you can only conclude with 100% certainty that the coin is unfair.
The only thing that changed is the understanding of the population it was drawn from.
Okay. So why is this a problem? For the simple reason that the coin labelled 0.5 exists.
If you reflect on the plight of coin 0.5 you will understand why it is very wrong to talk about the bag from which it was drawn or how.
Exercise: generalize this result for finite cases.
References: