Whats the probability that?

Question

A computer program generates a sequence of 5 independent random digits. Find the probability that exactly 2 of the digits are 9’s.

Answer 1

You have 10 choices for each digit (0-9), so the number of possible strings of 5 digits is:

10 × 10 × 10 × 10 × 10 = 100,000

And then you want 2 digits to be 9s, and the other three to be anything else.

You have 5 places to put the 2 9s, and both the 9s are the same (that is, the first 9 in position 1 and the second 9 in position 2 is the same as the first 9 in position 2 and the second in position 1), so there are C(5, 2) ways to pick the locations of the 9s, and then you have 8 choices for each other digits.

5 × 4 / 2 × 8 × 8 × 8 = 5,120 strings of length 5 with exactly 2 9s.

(If you can't see that, look at this. Each 1 represents the location of a 9, and each 8 represents every digit that is NOT 9:

1 × 1 × 8 × 8 × 8 +
1 × 8 × 1 × 8 × 8 +
1 × 8 × 8 × 1 × 8 +
1 × 8 × 8 × 8 × 1 +
8 × 1 × 1 × 8 × 8 +
8 × 1 × 8 × 1 × 8 +
8 × 1 × 8 × 8 × 1 +
8 × 8 × 1 × 1 × 8 +
8 × 8 × 1 × 8 × 1 +
8 × 8 × 8 × 1 × 1 =
5,120)

So the probability is 5,120/100,000 = 5.12%