Combinatorics & probability density

[Cathr]

Suppose we have two boxes, each containing three types of balls. On each ball there's written a number:
First box: 1, 2, 3
Second box: 4, 5, 6
We don't know how many balls of each type there are, but we know the probability of taking out a specific one, so that we can make a graph showing the discrete probability density of each ball.
Knowing the probability of each number, can we calculate the probability of combinations of numbers?
Let's say if we take a random number from each box, how can we calculate the probability of the combination?
And what if, for example, we take out 3 balls from the first box and 3 from the other at once? (Here we must consider two cases: when the numbers may repeat themselves and when they are all different).

Thanks for any help!

 

[StoneTemplePython]

Are you sampling with or without replacement from each box? If without replacement, not knowing the underlying number of balls is going to be a problem (unless, of course, you can assure yourself it is sufficiently large and hence can be approximated by sampling with replacement...)

There's a lot of ways of attacking this when sampling with replacement -- drawing a directed graph of the process to model it as a finite state Markov chain main be enlightening.

We don't know how many balls of each type there are, but we know the probability of taking out a specific one, so that we can make a graph showing the discrete probability density of each ball.

Loosely speaking there is no such thing as a "discrete probability density". You can plot a PMF. Or you could work with a CDF here, but I don't think that's what you're talking about.

 

[Cathr]

Are you sampling with or without replacement from each box? If without replacement, not knowing the underlying number of balls is going to be a problem (unless, of course, you can assure yourself it is sufficiently large and hence can be approximated by sampling with replacement...)

There's a lot of ways of attacking this when sampling with replacement -- drawing a directed graph of the process to model it as a finite state Markov chain main be enlightening.

Loosely speaking there is no such thing as a "discrete probability density". You can plot a PMF. Or you could work with a CDF here, but I don't think that's what you're talking about.

Thanks for your response!

It can be approximated to sampling with replacement - we may imagine that boxes are infinitely large. What interests me is trying to find how the probability of picking each number separately can influence the probability of combinations of the numbers. Is there a formula that I can use? Can a finite state Markov chain model be applied here? I am not familiar with it at all, but if it's the case I will read more about it.

Also I spotted a mistake in the last sentence of the problem, when I considered that we pick 3 balls out of 3 from each box and they are all different. Obviously there's just one possible case, so I should have said that we pick 2 balls out of 3, here we may have more cases. Then the number of different balls in each box can be increased - to 10, 100 but that's more complicated because we would have more combinations.