Jar and Ball Probability Distribution

Question

If I have 8 jars, each jar contains 5 unique ball types. However, I know that I have 20 unique ball types out there. So, I have balls labelled from B1, B2, B3, ...B20 to put into 5 jars.

Let's say

Jar 1 : B1, B3, B4, B7, B12 (5 unique balls)

jar 2 : B9, B7, B11, B15, B18 (5 unique balls)

jar 3 : B20, B1, B5, B17, B13 (5 unique balls)

jar 4 : B4, B5, B6, B3, B17 (5 unique balls)

jar 5 : B3, B10, B2, B14, B16 (5 unique balls)

and so on...

My question :

1. What is the probability to observe one particular ball type in a jar ?

2. What is the probability to observe one particular ball type in at least n jars ?

Is there a well-known probability distribution or combinatorial method that might relate to this ?

Thanks

Answer 1

Question 1:

There are ${ 20 \choose 5 }$ ways to choose $5$ distinct ball types to a jar. For ball type $i$, there are ${ 19 \choose 4}$ ways to choose $5$ ball types such that $i$ is one of them. Thus the probability is $$ p = {19 \choose 4}/{20 \choose 5} $$

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Question 2:

The probability to observe ball type $i$ in a jar is $p$, as shown above. The probability to observe $i$ occurring in $0 \leq n \leq 8$ jars of the $8$ jars is: $$ {8 \choose n} p^n (1-p)^{8-n} $$ which is a binomial distribution.

Thus the probability of occurring in at least $n$ jars is: $$ \sum_{j = n}^8 {8 \choose j}p^j(1-p)^{8-j} $$