Basic probability problem

Here's one from one of my problem solving books:

Let x be an integer randomly selected, such that What is the probability that x is divisible by 7, 11, or both?

This is basically a counting problem, as most probability problems are: We count the total amount of numbers which satisfy the conditions, and divide by the total amount of numbers. So we're looking for the amount of numbers from 1 to 500 which are divisible by 7 or 11 or both. So lets start with 7:
Largest multiple of 7 not greater than 500 is 497 (7 * 71), so there are 71 numbers divisible by 7.
Now for 11:
Largest multiple of 11 not greater than 500 is 495 (11 * 45), so there are 45 numbers divisible by 11.

Now, it would be a mistake to add these together and divide by 500 to say that the probability is . Why?
The reason this is wrong is because we counted the numbers which are divisible by 7 and 11 twice: once when we counted numbers divisible by 7, and again when we counted numbers divisible by 11. To compensate, we have to subtract the amount of numbers divisible by both, so we only count them once.

7 and 11 are prime, so the only numbers which are divisible by both are multiples of 77 (7 * 11):
Largest multiple of 77 not greater than 500 is 462 (77 * 6), so there are 6 numbers divisible by both 7 and 11.
Now we can find our probability: