Bob plays a game where, for some number $n$, he chooses a random integer between 0 and $n-1$, inclusive. If Bob plays this game for each of the first four prime numbers, what is the probability that the sum of the numbers he gets is greater than 0?
The first four primes are 2, 3, 5, and 7. The only way for the sum of the numbers Bob gets to not be greater than 0 is if he gets a 0 each time he plays the game. The chance that Bob will get a 0 each time he plays is $\frac{1}{2}\cdot \frac{1}{3} \cdot \frac{1}{5} \cdot \frac{1}{7}=\frac{1}{210}$. Therefore, the probability that Bob will not get all 0's is $1-\frac{1}{210}=\boxed{\frac{209}{210}}$.