Given some arbitrarily large $a$, what is the probability that this number is a prime number?
My attempt involves seeing that for $a$ to be prime, then it must not have a factor $N$ in the following range:
Which helps make the problem somewhat easier.
Then, I noted all the possible prime numbers in the range for needed factors.
I will call them $p_1,p_2,p_3,\dots p_i$.
I think, if I am correct, that the probability that $a$ will be prime is as follows:
Of course, this requires knowledge on all of the primes in the given range, which I want to avoid.
So my questions are, is my probability correct? And is there another formula that finds the probability of a number $a$ being prime without the need of anything other than the given number $a$?
Note that the probability of a number being prime is not the same as the question whether or not a number is prime.
Ok, according to the comments, I should probably clarify things. If this helps, I wish to find the probability that a given number is prime, and I want to find the individual probability of each natural number in terms of itself.
That is, the probability $P_a$ that $a$ will be prime is given as:
Where we apply some operation to find the probability that $a$ is prime.
To clarify what I mean by probability of a number being prime, I mean that given a number prime, how probable is it to be prime? (sorry for confusions).
For example, can we agree that $10$ is more likely to be prime than $100$? It is apparent that numbers that are smaller are more often prime numbers than large numbers.
So, it might be interesting to notice that:
Still confused? Or perhaps this question is just unanswerable.
So I ask a second question, one that may be more clear.
In a given range, $a\le N\le b$, how many prime numbers $N$ are expected to exist in that range? I think that this second question is clearer, but if it still raises concerns, do inform me.