Question

A fair die is rolled repeatedly. Find the expected number of rolls until all 6 faces appear.

Answer 1

The chance of rolling a number you haven't yet rolled when you start off is 1, as any number would work. Once you've rolled this number, your chance of rolling a number you haven't yet rolled is 5/6. Continuing in this manner, after you've rolled n different numbers the chance of rolling one you haven't yet rolled is (6−n)/6.

You can figure out the mean time it takes for a result of probability p to appear with a simple formula: 1/p. Furthermore, the mean time it takes for multiple results to appear is the sum of the mean times for each individual result to occur.

This allows us to calculate the mean time required to roll every number:

t=1/1+6/5+6/4+6/3+6/2+6/1

=1+12/10+15/10+2+3+6

=12+27/10

=14.7