Probability of Assigning r Balls in n Urns with m Urns Containing k Balls

[Yankel]

Hello all,

I am trying to solve this problem:

r balls are randomly assigned into n urns. The assignment is random and the balls are cannot be distinguished. What is the probability that exactly m urns will contain exactly k balls each ?

I know that the probability of each ball to be in each urn is 1/n.
I addition I have (nCm) ways to choose which are the urns to be filled with k balls.
Then I have r−km balls to distribute in the n−m remaining urns.

I do not know how to proceed. Can anyone solve this difficult problem ?

My current attempt is:

\[\frac{\binom{n}{m}\binom{r-km+n-m-1}{r-km}}{\binom{r+n-1}{r}}\]

It must be wrong...

 

[mmwave]

Hello,

Thank you for bringing this interesting problem to our attention. I will try my best to provide a solution to this problem.

To find the probability that exactly m urns will contain exactly k balls each, we can approach this problem using combinations. Let's break down the problem into smaller parts:

1. Probability of selecting m urns out of n urns: We can do this in \(\binom{n}{m}\) ways.

2. Probability of placing k balls in each of the selected m urns: Since the assignment is random and the balls are indistinguishable, we can think of this as placing k identical balls in m distinct urns. This can be done in \(\binom{r}{k}\) ways.

3. Probability of placing the remaining r-km balls in the remaining n-m urns: We can think of this as placing r-km identical balls in n-m distinct urns. This can be done in \(\binom{r-km+n-m}{r-km}\) ways.

Now, to find the total number of ways of distributing r balls into n urns, we can use the stars and bars method. This method states that if we have r identical balls and n distinct urns, the number of ways to distribute the balls into the urns is \(\binom{r+n-1}{r}\).

Therefore, the probability of exactly m urns containing exactly k balls each can be calculated as:
\[\frac{\binom{n}{m}\binom{r}{k}\binom{r-km+n-m}{r-km}}{\binom{r+n-1}{r}}\]

I hope this helps in solving the problem. Let me know if you have any further questions or if you would like me to explain any step in more detail.