I am not a mathematician but enjoy studying the primes as a mechanical system. Regrettably I don't have the tools (and perhaps the intellect) for some jobs.

I've spent some time trying to calculate the probability of a number being a product of each successive prime as the number line is progressively sieved, and have decided it cannot be done. But maybe this is just ignorance.

It is simple for 2,3, and 5.

Sieve out the products of 2 and 1/3 of the remainder will be products of 3.

Sieve out the products of 3 and 1/5 of the remainder will be products of 5.

Sieve out the products of 5 and about 1/7.5 of the remainder...

From here on the proportion become increasingly difficult to calculate.

Is this calculation possible? Or must it always be a statistical approximation?