### Brain Teasers

# Prime Probability

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

Given any positive whole number, what is the probability it is prime?

### Hint

Use Prime Number Theorem.### Answer

Zero.Basically, all we need to do is divide the number of prime numbers by the number of positive whole numbers.

But yes, they are both infinite, so we should consider taking the limits.

As in the Prime Number Theorem, PI(x) denotes the number of primes less than x.

We can rearrange this directly to give...

lim[x->infinity] { PI(x)/x} = lim[x->infinity] {1/log x}

Since the right hand side is 0, this gives us 0 for the answer.

(A probability of 0 does not necessarily mean impossible. These situations usually arise when infinite quantities are involved.)

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