Premise
Lets say $S$ is the set of all unique items $s$ from which currently have been split off into $H_i$.
$H_1$ $H_2$ $H_3$ are sets that already contain $n$ specific elements $s$ from $S$. We don't know which ones they are. If an element is in $H_1$ then it can't be in $H_2$ or $H_3$.
$P_1$, $P_2$, and $P_3$ are subsets of $S$. Unlike $H_1$,...,$H_3$, these subsets can possibly overlap with each other.
$P_i$ contains all elements from $S$ that $H_i$ can possibly contain.
Our Knowns
- For all $s$ ∈ $S$, we know which $P_i$ is it also in
- We know the size of each $H_i$
- We know the size of each $P_i$
Question
Given $P_1$, $P_2$, and $P_3$ and the elements $s$ ∈ $S$ that they contain, and given the sizes of $H_1$, $H_2$, and $H_3$ how can we determine the probability for each $s$ ∈ $S$ of it belonging to $H_1$, $H_2$, and $H_3$.
My understanding so far
I know that in cases where $P_i$ and $H_i$ are the same size, then $s$ ∈ $S$ have 100% chance of being in $H_i$. But in cases where the size of $P_i$ is larger than $H_i$ for i ∈ {1,2,3} and the size of the $H$ sets are different, how can I determine the probabilities?
$H_1$ size = 4 {2,4,15,13} $H_2$ size = 5 {6,8,10,12,14} $H_3$ size = 6 {1,3,5,7,9,11}
$P_1$ size = {1,2,3,4,5,6,7,8,13,14,15}
$P_2$ size = {5,6,7,8,9,10,12,13,14,15}
$P_3$ size = {1,2,3,4,5,7,8,9,11}
While we're not supposed to know $H_i$, I put them there just for us, but we don't have that information.
We know $H_1$ is 4$C$11
We know $H_2$ is 5$C$10
We know $H_3$ is 6$C$9
In this example, how can I deduce each numbers' chance of belonging in H1-3, and how can I generalize that for all combination of hand sizes and $P_i$s.