# I Hypergeometric distribution with different distributions

1. Mar 17, 2016

### Hex5f

Hello,
For this type of question:

There are 5 green and 45 red marbles in the urn. Standing next to the urn, you close your eyes and draw 10 marbles without replacement. What is the probability that exactly 4 of the 10 are green?

I understand that I can use Hypergeometric distribution, which takes into account the changing probability of the balls after each draw. But how can answer the same question when the green balls are distributed with some distribution (exponential for example) ?
I've been thinking that I may calculate the expected value of the green marbles and then calculate this as before when the population of the green marbles is the expected value. Would it be correct ?

Thanks

2. Mar 17, 2016

### andrewkirk

Your original situation is about a probability. But in the end you seem to be asking about an expected value, which is not the same question. What is it an expected value of? For instance, does each ball have a real number written on it, randomly drawn from some distribution? The distinction between red and green marbles will be irrelevant unless they have different distributions for the numbers written on them.

3. Mar 19, 2016

### Hex5f

Are you saying that if all marbles are identical among each group it does not matter how each group distributes ?

4. Mar 19, 2016

### andrewkirk

I don't know what you mean by 'how each group distributes'. Perhaps you mean the way of determining the probability that the next drawn marble will be green, given that there remain g green and r red marbles in the urn.

The standard assumption is that that probability is g/(g+r), because that's the simplest situation and seems to match most real-life urn-drawing activities.

But there could be ways of changing that probability, for instance by making the green ones heavier so they are more likely to be at the bottom of the urn, or making them sticky, or some other disturbance of the system. In that case the probability would no longer be g/(g+r) and the hypergeometric distribution would not longer be applicable. A new distribution would need to be worked out based on the specifics of what the new probability of drawing a green marble was.

5. Mar 21, 2016

### Hex5f

That's exactly what I'm trying to calculate. So, would it be correct to calculate the expected value of green balls, I mean the expectation of the number of green balls ones find when drawing a handful of marbles and then use this value as the total amount of green balls in hypergeometric distribution calculation ?

Else, would it be correct sampling from the distribution of green marbles for each selection of a green marble ?

If not, what is the correct direction for calculating this ?

Thank you