Understanding Hyper-geometric Distribution with excel:

Assume that a box contains 10 Plugs of which 6 are good and the rest defective. An operator picks 5 Plugs at random from the 10, and is interested in the number of good Plugs picked. 

Let X denote the number of good Plugs picked. If in a Population of size N contains S successes and (N - S ) failures, and a random sample of size n is drawn from the pool, the number of successes X in the sample follows then Hyper-geomatirc Random Variable can be defined as:

$$Hypergeomatric (x,N,n,k) = \frac{\binom{k}{x}\binom{N-k}{n-x}}{\binom{N}{n}}$$

Lets understand it with an example, for the case described in above paragraph, where N = 10, S = 6 , n = 5 , x = 2.

The Hyper-geomatric Distribution can be use to calculate the chances of getting 02 defective plugs in the sample by this:
 
$$Hypergeomatric (x,N,n,k) = \frac{\binom{5}{2}\binom{10-5}{6-2}}{\binom{10}{6}}$$ 
..This will give you exact probability of 02 Plugs.

For at most 02 Plugs i.e. at max 02 plugs will be defected:

$$Hypergeomatric (x,N,n,k) = \sum_{x=0}^{x=2}\frac{\binom{k}{x}\binom{N-k}{n-x}}{\binom{N}{n}}$$ 
 For atleast 02 Plugs i.e. at max 02 plugs will be defected:
 
$$Hypergeomatric (x,N,n,k) = 1-\sum_{x=0}^{x=2}\frac{\binom{k}{x}\binom{N-k}{n-x}}{\binom{N}{n}}$$

All this can easily be computed using HYPGEOMDIST function of Excel. This takes follwoing arguments:   

HYPGEOMDIST(sample_s,number_sample,population_s,number_population)


Now using the Excel's Hypergeomatric Function the computation is easily done. Substituteing Values will give you the same result as you have calculated through the conventional formulas.

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