A number is chosen uniformly at random from among the positive integers less than 10^8. Given that the sum of the digits of the number is 9, what is the probability that the number is prime?

If the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

And if a number is divisible by 9 then it is not prime.

So given that the sum of the digits is 9, the probability that the number is prime is 0.