This is a special case of the Binomial distribution only that it is without replacement.
Properties
The following are characteristics of hypergeometric experiments:
- Samples are taken from two groups.
- The interest is in the first group.
- Sampling from the combined groups is done without replacement.
- Each sample draw is not independent. This is to say the probability of each sample pick is dependent on the previous sample pick(s).
- It is important to bear in mind that these are not Bernoulli Trials.
Characteristics
- The hypergeometric distribution consists of a finite number of trials.
- In each trial, the probability of success differs.
Example
Let’s say that you have $1,100 in your pocket. Furthermore, $800 is in 100-dollar bills and the rest is in 50-dollar bills.You randomly draw 5 bills without replacing. Find the probability that you will draw exactly 3 50-dollar bills.
Answer
The number of 100-dollar bills is $800/$100 = 8
The number of 50-dollar bills is $(1100-800)/$50=$300/$50 = 6
In total, you have 8 + 6 = 14 bills.
Hence our parameters take the following values;
N = 14; Population size
k = 6; Number of successes in the population
n = 5; Sample size
x = 3; Desired number of successes in the sample
We now substitute the above parameters into the hypergeometric formula as shown below;
P = [ 6C3 ] [ 14-6C5-3 ] / [ 14C5 ]
P = (20*28)/2002
P = 0.2797
We find that the probability of drawing exactly 3 50-dollar bills from your pocket given the above parameters is 0.2797.
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