Question

What is the probability that a 7-digit phone number contains at least one 8? (Repetition of numbers and lead zero are allowed).

Answer 1

Let's compute the probability that a 7-digit phone number does not contain 8s, at all.

This means that, for each of the 7 digits, we have 9 choices (the ten digits except 8).

So, there are $7^9$ 7-digit phone numbers that don't contain 8s.

Note that the total number of 7-digit phone numbers is $7^{10}$, because we have 10 choices (the ten digits) for each of the seven slots.

So, the probability of finding a phone number with no 8s is

$\dfrac{7^9}{7^{10}} = \dfrac{1}{7}$

This implies that the probability that a 7-digit phone number contains at least one 8 is the complementary probability

$1-\dfrac{1}{7}=\dfrac{6}{7}$