8. In a certain school, there are 5000 students. Each student is assigned an ID number from 0001 to 5000 . No two students can have the same ID number. If a student is selected uniformly at random, what is the probability that the ID number of the student does not contain any 2 s among its digits?
Answer: $\frac{729}{1250}$
Solution: The number of ID numbers in the range 0001 to 5000 which do not contain any 2 s is given by $4 \times 9 \times 9+1-1=2916$. For the thousands digit, we can use a number from 0 to 4 and for the rest we can use any digit except 2. We add one since 5000 was not counted and we subtract one since there is no ID number of 0000 .
Thus, the probability that the student ID number does not contain any 2 among its digits is $\frac{2916}{5000}=\frac{729}{1250}$.