TITLE: Probability for specific outcomes in fair die rolls
QUESTION [7 upvotes]: We roll a fair die n times. We are looking for the maximum n, for which, the probability that the maximum value that appears in all outcomes is 5, is >24%.

At the beginning I thought that the probability would be $(\frac{5}{6})^n$, that is, the probability for one particular number not to be included in the outcomes (6, in our case). But then I figured out that we could have also cases where some other numbers are missing, but 5 is still the largest of those that appear. Any ideas?

REPLY [6 votes]: Let's find the total number of possibilities for the dice rolls. For every dice roll, you have 5 possibilities. This gives $5^n$ possibilities. However, you need to remove the possibility that no die is a five. There are $4^n$ possibilities of no 5 and no 6. This is out of a total number of possible ways to roll $n$ dice of $6^n$. So, the probability is:
$$\dfrac{5^n-4^n}{6^n}$$
Now, you are looking to solve:
$$\dfrac{5^n-4^n}{6^n}= .24$$
Plugging this into Wolframalpha gives about 6.27, so 6 is the maximum $n$.