For each out someone has, they are approximately 2% to hit it on a given street. So with 15 outs, he has about a 30% chance to hit one of them on the turn, and if he fails to do that, he has about a 30% chance to hit it on the river. That means he has about a 60% chance to hit one, the other, or both.
That's a bit of an over-estimate, as that math is rounding, and the more outs you have the more rounding error you get, but it's still well over 50%.
54.12%, to be exact.