What is the probability that a randomly chosen positive three-digit integer is a multiple of $7$? What is the probability that a randomly chosen positive
three-digit integer is a multiple of $7$?

Is my answer right?:

$$\frac{100}{7} = 14 , \qquad \frac{999}{7} = 142$$

Then there are $142 - 14 = 128$ numbers that are multiples of $7$.

Then the probability is: $\frac{128}{900}$.
Yes, except that:


I would replace $\frac{100}{7}$ with $\frac{99}{7}$ (to see why this is important consider the analogous question asking for how many multiples of $7$ there are in $\{700, \ldots, 999\}$), and
the quotients aren't quite right as written, but we can repair them with floor notation, e.g., $\lfloor\frac{999}{7}\rfloor = 142$.
Find the lower bound and upper bound that is a multiple of 7! so for a three digit integer that is a multiple of 7, 100 is the lowest possible three digit integer and 999 is the highest, however, neither of these are a multiple of 7!
So starting at 100 find a multiple of 7! (105) is the product of (15*7), that is your lower bound. Now find your upper bound,starting at 999; you find, (994) which is a product of (142*7)! Now use a THEOREM (IF m and n are integers, and m &lt;= n,then there are n-m +1 integers from m to n, inclusive). This means take the integer 142 from the product of 142*7 from your upper bound, and subtract that from your lower bound product which is the integer 15 from the product 15*7. You will get your event (E) so E = 142 - 15 + 1 = 128. So E = 128, that is how many three digit integers are a multiple of 7! So now take your E and put that over your sample space (S) where S = all possible outcomes in a random outcomes. so (s) = 9 * 10 * 10 where the first digit of your three digit number is a natural number between 1-9, the second digit can be any natural number from 0 to 9, and the third digit number can also be a natural number between 0 and 9. So you get (S) = 9 * 10 * 10 = 900.  So E/S gives you your ratio or probability of number of three digit integers that are a multiple of 7.  You can use this for any three digit multiple, or four digit multiple ... (N) multiple where N is a non negative  member of Z.