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My friend who is doing elementary mathematics asked me, about prime number theorem. I tried to tell him in a simple language, that prime number theorem sort of tells you that the given sufficently large number $N$, the probability that a randomly chosen natural number from the set $\{1,2,3,4,5 \dots N-1, N\}$ being a prime number is $\frac{1}{\log N}$. He thought for a while, and immediately followed up with another question. What is the probability that a random number $N$ is a prime. I am not sure how to answer this question. Can anyone help me with this. I feel we need the definition of probability space without which we cannot answer this question. Thank You!

GraduateStudent
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    See https://en.wikipedia.org/wiki/Natural_density – lhf Nov 30 '24 at 16:12
  • $\log x$ is almost constant over most of the interval, so the probability of $N$ being prime is also $1/\log N$ – Empy2 Nov 30 '24 at 16:38
  • The problem is not about prime numbers. The problem is : Whe we say 'select a random integer' this is not clear, what does it mean ? Nothing. You need to change this process 'Select a random integer' by something else, correctly defined. And then you will have the possibility to answer further questions, like probability to obtain a prime number. – Lourrran Nov 30 '24 at 17:14
  • its not about selecting a 'random' number. Its about selecting a number and then wondering what is the probability that it is prime depending on the number we selected. The problem is we do not know the probability space. We cannot answer untill it is clarified. For example, we can define something like there are two possibilites, either a number is prime or it is not prime, in which case the odds are even. However the problem my friend asked is not answerable unless he specifies the probability space. – GraduateStudent Dec 01 '24 at 12:26
  • even I do not know what I am saying, everything seems too vague to answer properly – GraduateStudent Dec 01 '24 at 12:27

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You are absolutely correct $-$ the question is meaningless unless your friend specifies a probability distribution on the natural numbers. This is because there is no uniform distribution on the whole of $\Bbb N$. You can see this quite easily: if all natural numbers had the same probability $p$, then both $p=0$ and $p\ne 0$ would lead to contradictions.

TonyK
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