Well, sure that works on a national basis for the U.S., but how about on a worldwide basis (I bet it's quite similar though)?
A quick search yields that the current world population is roughly 6,959,054,865 (Sep 1. estimate), and the death rate is roughly the same: 8 deaths/1,000 population.
Plugging those numbers into your method, the number of deaths per year is
6,959,054,865 * (8 / 1 000) = 55,672,438.9
And the number of deaths per day (using 365.25 to account for leap years):
55,672,438.9 / 365.25 = 152,422.831
And, finally, everyone's individual chance of dying:
152,422.831 / 6,959,054,865 = 0.0219028063
That's roughly a one in 5000 chance.