Expected value of highest dice rolled
You roll $N$ six-sided dice simultaneously. Your score is the highest number rolled. If you play this game many times, what is the expected value (mean) of your score?
- A positive integer $N$.
- Your code must work for inputs up to and including 10, but may crash, error, or give incorrect output for larger inputs.
- The expected value (the mean value) of the highest individual dice result when $N$ six-sided dice (with face values 1, 2, 3, 4, 5, 6) are rolled simultaneously.
- For inputs up to and including 10, your output is valid if rounding it to 6 decimal places results in the output shown in the test cases.
Note that this means that if you find an incorrect algorithm that happens to give the correct result when rounded to 6 decimal places for inputs from 1 to 10, that is still a valid entry.
Test cases are in the format
input : output.
1 : 3.500000 2 : 4.472222 3 : 4.958333 4 : 5.244599 5 : 5.430941 6 : 5.560292 7 : 5.654117 8 : 5.724354 9 : 5.778177 10 : 5.820159
This is a code golf challenge. Your score is the number of bytes in your code.
Explanations are optional, but I'm more likely to upvote answers that have one.