# Does any Probability Distribution Functions Exist that can Model this Problem?

I invented these two problems which can might correspond to a probability distribution - but I am not sure which ones.

Case 1: Suppose there are 5 "hats" : the first "hat" has 10 red balls, the second "hat" has 10 blue balls", the third "hat" has 10 green balls, the fourth "hat" has 10 grey balls and the fifth "hat" has 10 pink balls.

You first select a random number of hats (each hat has a equal probability of being selected) - and then from these randomly selected hats, you select a random number of balls (within a given hat, each ball has an equal probability of being selected).

Example:

• Iteration 1: First Hat, Fourth Hat - 5 red balls, 2 grey balls
• Iteration 2: First Hat, Second Hat, Fifth Hat - 8 red balls, 1 blue ball, 6 pink balls

Case 2: Suppose there are 5 "hats" : the first "hat" has 10 red balls, the second "hat" has 10 blue balls", the third "hat" has 10 green balls, the fourth "hat" has 10 grey balls and the fifth "hat" has 10 pink balls. There are also 5 "bins" containing 1000 grams of sand.

You first select a random number of hats (each hat has a equal probability of being selected) - and then from these randomly selected hats, you select a random number of balls (within a given hat, each ball has an equal probability of being selected). Second, you select a random number of "bins" and scoop out a random weight of sand from each "bin".

Example:

• Iteration 1: Third Hat, First Bin, Fifth Bin : 2 green balls, 231 grams of sand, 401 grams of sand

• Iteration 2: First Hat, Fourth Hat, Fifth Hat, Second Bin, Fifth Bin : 5 red balls, 1 grey ball, 0 pink balls, 31 grams of sand, 891 grams of sand.

My Question: Are there any probability distributions that correspond to Case 1 and Case 2?

For instance, I am guessing that Case 1 corresponds to 2 different hypergeometric distributions and Case 2 corresponds to 2 different hypergeometric distributions as well as a normal distribution?