/sci/ - Science & Math

>>15353202
Apparently the hardest concept of them all is that there is no hardest concept. There is no scala naturae in the world and isn't a puzzle to be solved by humans. The 99.999(9)% of the universe exists for it's own sake and us being able to intelectually engage with it is just a coincidence.

Second place would be probability because it's unnatural to us.

When ChatGPT was trained without much human supervision it showed very good grasp of probability but was pretty bad at other stuff. When curated by human evaluations it became much better at everything EXCEPT statistics and probability. They got noticeably worse.

It makes sence. Our most powerful ability is recognizing patterns. The whole IQ test is about it fundamentally. The frontal lobe which is considered to be the most unique in humans if associative in it's function. It's also full of hard settings that are baked in and are hard to remove. They are called cognitive distortions, there a list on Wikipedia.
One of them is seeing pattterns that are not there.
Which is why it's so hard to understand that after you get 100 tails the odds of it being heads is still 50%. We just can't do it without strong abstraction and rationalising.

>TL;DR: How do you approach binomial probability when the sets are of infinite size?

For example, say you flip a fair coin an infinite number of times, representing every head with a "1" and every tail with a "0". The results will be of the form:

[00000000...00000000]
[00000000...00000001]
[00000000...00000011]
...
[11111111...11111100]
[11111111...11111110]
[11111111...11111111]

How would you calculate the probability that your resultant set will contain, let's say, exactly half heads and half tails. There are many sets for which this is true, such as:

[10101010...10101010]
[00110011...00110011]
[11111111...00000000]
And so on...

How do you calculate the probability that your coin flip experiment produces exactly half heads and exactly half tails? Is it probability zero? Or is this calculation impossible?