A fair die is thrown 20 times, find prob. that all sides appear at least once

Hi, I have the following problem which I don't know how to solve:

"A fair die is thrown 20 times. Find the probability that all of the six numbers appear at least once."

I think the number of possible combinations is 6^20, but I don't know how to work out the numerator... I know 14.7 is the average number of times that the die needs to be thrown so that all sides appear at least once (1+6/5+6/4+6/3+6/2+6/1=147/10=14.7), but I don't know if I have to use this, and if I do, I don't know how (Worried). Thanks a lot for your help!

Re: A fair die is thrown 20 times, find prob. that all sides appear at least once

Quote:

Originally Posted by

**juanma101285** "A fair die is thrown 20 times. Find the probability that all of the six numbers appear at least once."

**You will need to apply the inclusion/exclusion principle.**

You need to find the probability that at least one number does no appear and subtract that from 1.

When a fair die is thrown 20 times the probability **that 1 will not appear** is $\displaystyle \left( {\frac{5}{6}} \right)^{20} $

When a fair die is thrown 20 times the probability **that 1 & 2 will not appear** is $\displaystyle \left( {\frac{4}{6}} \right)^{20} $

etc.

Re: A fair die is thrown 20 times, find prob. that all sides appear at least once