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

1. ## 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 . Thanks a lot for your help!

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

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

Thanks so much!

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