I need to determine a probability for some work related to my Master Thesis. It may seem really easy but I've been trying for the last two days and I couldn't get any solution:
Imagine that we roll n times a 6-sided die. What's the probability of getting each result (1, 2, 3, 4, 5, 6) at least once?
Note: n should be greater than 6 (or equal), to have a probability greater than zero.
Given the number of times we get every result, we can calculate the probability using a multinomial distribution. For n=6 or even for n=7 the solutions are easy, but as n gets bigger the possibilities grow really fast, so it doesn't seem feasible to use combinations of multinomials. Since the problem seems really easy, I was wondering if there is a simple solution for this.
Thanks a lot.
Just for curiosity, is that formula the probability mass function of any known distribution? Does it have any name? I am that kind of person that likes to understand everything, but looks like this formula may be out of my possibilities.
This probability I wanted to get, if I'm not wrong, can also be obtained as a summatory of all the probabilities we can get with the multinomial distribution:
such as and
The number of vectors that can be written following that condition grows a lot when n grows, so calculating the summatory of all those probabilities may be not feasible for large values of n, that's why I was asking for a easier solution.
However, I can't seem to find the relation between your formula and the multinomial distribution. Actually, is that what really confuses me.
which is your result.
Thank you very much!