Hypergeometric distribution with different distributions

In summary, if all the marbles are the same color, then the probability of drawing a green marble is g/(g+r). If there are some green and some red marbles, then the probability of drawing a green marble is (g+r)/2. If all the marbles are different colors, then the probability of drawing a green marble is (g+r)/n, where n is the number of different colors.
  • #1
Hex5f
3
0
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
 
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  • #2
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
Are you saying that if all marbles are identical among each group it does not matter how each group distributes ?
 
  • #4
Hex5f said:
Are you saying that if all marbles are identical among each group it does not matter how each group distributes ?
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
andrewkirk said:
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.

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
 

1. What is a hypergeometric distribution?

A hypergeometric distribution is a probability distribution that describes the probability of obtaining a certain number of successes in a fixed number of samples from a population of a fixed size that contains a specific number of successes and failures.

2. How is a hypergeometric distribution different from a binomial distribution?

A binomial distribution describes the probability of obtaining a certain number of successes in a fixed number of independent trials, where each trial has the same probability of success. In contrast, a hypergeometric distribution accounts for the fact that the probability of success may change with each trial, as the population size decreases.

3. Can a hypergeometric distribution have different distributions?

Yes, a hypergeometric distribution can have different distributions depending on the specific parameters used. For example, the distribution may change if the population size or the number of successes in the population changes.

4. How is a hypergeometric distribution used in real-world applications?

A hypergeometric distribution can be used to model situations where a sample is taken from a population without replacement. This distribution is commonly used in quality control, market research, and election polling.

5. What are the assumptions of a hypergeometric distribution?

The assumptions of a hypergeometric distribution include a fixed population size, a fixed number of successes and failures in the population, and a fixed sample size. Additionally, the samples must be drawn without replacement and each sample must have the same probability of success.

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