What is the probability that a 8-digit phone number contains at least one 5? (Repetition of numbers and lead zero are allowed).

To find the probability that an 8-digit phone number contains at least one 5, we need to calculate the total number of possible 8-digit numbers and the number of those numbers that do not contain a 5.

To calculate the total number of possible 8-digit numbers, we need to consider that the phone number can include any digit from 0 to 9 in each of the 8 positions. Since repetition of numbers is allowed, we have 10 options (0-9) for each digit position.
So the total number of possible 8-digit numbers is 10^8.

Now, let's calculate the number of 8-digit numbers that do not contain a 5. Since the numbers can include any digit from 0 to 9, except 5, we have 9 options (0-4 and 6-9) for each digit position.
So the number of 8-digit numbers that do not contain a 5 is 9^8.

Finally, the probability that an 8-digit phone number contains at least one 5 is equal to 1 minus the probability that it does not contain a 5.
So the probability is given by:

1 - (number of 8-digit numbers that do not contain a 5) / (total number of possible 8-digit numbers)

= 1 - (9^8) / (10^8)

To find the actual value, we can calculate this expression using a calculator or math software. The result is approximately 0.6513, which means that there is a 65.13% probability that an 8-digit phone number contains at least one 5.