Probability of getting every result after n tries

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.

Re: Probability of getting every result after n tries

Quote:

Originally Posted by

**brokenlugs** 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.

This part is easy.

That is the probability of each value appears at least once in trials

Quote:

Originally Posted by

**brokenlugs** 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.

I don't know exactly what that means.

Re: Probability of getting every result after n tries

Quote:

Originally Posted by

**Plato** This part is easy.

That is the probability of each value appears at least once in

trials

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?

Quote:

Originally Posted by

**Plato** I don't know exactly what that means.

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!

Re: Probability of getting every result after n tries

Quote:

Originally Posted by

**brokenlugs** 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.

Have you seen this webpage?

Re: Probability of getting every result after n tries

Re: Probability of getting every result after n tries

Quote:

Originally Posted by

**brokenlugs** 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

Re: Probability of getting every result after n tries