Hypergeometric distribution

Hypergeometric distribution is used to determine the probability of an event when drawing without replacement.

• $N$ is the population size
• $M$ is the number of success states in the population
• $n$ is the number of draws (i.e. quantity drawn in each trial)
• $k$ is the number of observed successes

The lottery model

The lottery model can be used to explain the hypergeometric distribution.

Example

How likely are 4 correct numbers in the lottery?

1. Total number of combinations

   The number of possible combinations can be determined using the binomial coefficient.
   
   ${49\choose 6}$ $=13,983,816$

2. Number of favorable events

   Now one imagines two groups: 6 winning balls and 43 blanks.
   
   First you determine the possibilities to choose from the 6 winning balls 4:
   ${6\choose 4}=15$
   
   Then the options to choose from 43 blanks 2:
   ${43\choose 2}=903$
   
   Multiplying both together gives the total number of ways to draw 4 winning balls and 2 blanks, regardless of the order:
   ${6\choose 4}\cdot{43\choose 2}$

3. Determine probability

   It is a Laplace experiment. The probability results from the number of possibilities for the event divided by the total number of all possible combinations:
   
   $P(X=4)=\frac{{6\choose 4}{43\choose 2}}{{49\choose 6}}$ $\approx0.001$
   
   You can see that this is a hypergeometric distribution with $n=6$, $k=4$, $M=6$ and $N=49$.