Hello,
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.
Wow, thank you very much!! This is what I was looking for! I wasn't even close to that. For a more general result, i.e. an experiment with j equiprobable possible results, I guess I just have to replace the 6 for a j and the 5 for a j-1. Am I wrong?
I meant that if I had to determine the probability of getting a 1 twice, a 2 three times, a 3 once, and so on (whichever combination I want), I could do that with a multinomial distribution, but looks like I was really far from getting a feasible result.
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.
Thank you!
Have you seen this webpage?
Yes, that's where I learned what a multinomial distribution is, and I think I understand it.
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.
The formula I gave you has little to do with multinomial distribution. Rather it is using inclusion/exclusion: read this page.
The new posting has probability
Thanks again for this information! I was able to get the same formula as you, so I understand everything now. The probability I was looking for was , where the event means that after rolling the dice n times, we didn't get the i-th side. Then, it's easy to calculate the probabilities of the intersections of the different and add them to the inclusion-exclusion formula:
which is your result.
Thank you very much!