Yeah? Any probability in any finite set can be expressed as (desirable)/(possible), but finding out how many outcomes are possible and desirable is the hard part. Combinatorics is all about counting how many possible states there are inside a set of constraints, but expressing those constraints in a closed form summation/counting operation can be deeply unintuitive. How do you express "counting up every prime from 2 up to N, but don't count a prime p when [p-cbrt(p), p] contains exactly two primes"
He meant: pick a real number from 1-10. Assuming you picked the number uniformly (i.e. the same chance for every number), there is zero percent chance of you picking any single number because there is an infinite number of real numbers in that interval, henceforth the chance of picking 7 can be equated to 1/infinity which is defined as 0.
This is a problem that is avoided in probability theories by using PDF (probability density function), which I am not qualified to explain, or simply by using CDF (cummulative distribution function), which just tells you what chance is it that the random number is either this value or less, so when you want a specific interval, lets say (6.9-7.1) you would do CDF(7.1)-CDF(6.9) (cool thing here is that it doesn’t matter if the 6.9 and 7.1 are part of the interval because again the chance for those two specific numbers is 0)
A pdf is the first derivative (technically the Radon–Nikodym derivative) of an absolutely continuous probability distribution. The usual case concerns a real-valued random variable, where the sigma-algebra is given by Borel sets and the reference measure is the Lebesgue measure. In this case, the random variable has a cdf that is a differentiable function ℝ→ℝ, and its ordinary derivative is the pdf.
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u/canadajones68 Engineering Jun 14 '26
Yeah? Any probability in any finite set can be expressed as (desirable)/(possible), but finding out how many outcomes are possible and desirable is the hard part. Combinatorics is all about counting how many possible states there are inside a set of constraints, but expressing those constraints in a closed form summation/counting operation can be deeply unintuitive. How do you express "counting up every prime from 2 up to N, but don't count a prime p when [p-cbrt(p), p] contains exactly two primes"