I am looking for a way on how to think about a probability problem. It feels like there's a way how to simplify the task.
So, here's example of the problem:
From a pool of $15$ balls ($1$ red, $2$ blue, $3$ green, $4$ yellow and $5$ black) a set of $5$ balls, at random, is selected and put in a bag. We don't know which $5$ balls are there.
Now, $2$ balls are at random picked from this set of $5$ balls, observed and put back with the other $3$.
Let's say we saw a blue and a yellow ball. Now i'd like to calculate probability of picking specific set of two balls (there are $14$ such sets).
One obvious approach I see is to list all the possible sets of $5$ balls what could be drawn from the initial pool of $15$ balls (there are $71$ such set), each with certain probability being drawn, $P(i)$; filter it to leave only those sets that contain at least one blue and one yellow ball (there are $25$ such sets), and then calculate the probabilities for each of the $14$ two-ball sets for each of those $25$ five-ball sets, weighing (multiplying) them by $P(i)$ and summing them up in $14$ sums.
The question I'd like to find out, is if there is simpler way to find the result avoiding so many iterations over specific sets.
Maybe there's some smart way how to think about those unseen $3$ balls in the bag, that would let me avoid iterating over the $25$ sets of $5$ balls that contain a blue and a yellow ball.