7.3. The Hypergeometric Probability Distribution:
Example:
Suppose there are 50 officers, 10 female officers and 40 male officers. Suppose 20 of
them will be promoted. Let X represent the number of female promotions. Then,
10  40 
  
 0  20 
P( X  0) 
 50 
 
 20 
# of combinatio ns for 0 female

10  40 
  
 1  19 
P( X  1) 
 50 
 
 20 

promotion (# of combinatio ns for 20 male promotions )
(# of combinatio ns for 20 promotions )
# of combinatio ns for 1 female
promotion (# of combinatio ns for 19 male promotions )
(# of combinatio ns for 20 promotions )

10  40 
 

 i  20  i 
P( X  i) 
 50 
 
 20 

# of combinatio ns for i female
promotion (# of combinatio ns for 20-i male promotions )
(# of combinatio ns for 20 promotions )

10  40 
  
10  10 
P( X  10) 
 50 
 
 20 

# of combinatio ns for 10 female
promotion (# of combinatio ns for 10 male promotions )
(# of combinatio ns for 20 promotions )
Therefore, the probability distribution function for X is
1
10  40 
 

i
20

i
 

P( X  i ) 
, i  0,1,  ,10.
 50 
 
 20 
Hypergeometric Probability Distribution:
There are N elements in the population, r elements in group 1 and the
other N-r elements in group 2. Suppose we select n elements from the
two groups and the random variable X represent the number of
elements selected from group 1. Then, the probability distribution
function for X is
 r  N  r 
 

i ni 
P( X  i )  f x (i )   
, 0  i  r.
N
 
 
n
r
  is the number of combinations as selecting i elements
i
N  r
 is the number of combinations as
from group 1 while 
 ni 
Note:
N
selecting n-i elements from group2.   is the total number of
n
 
combinations as selecting n elements from the two groups while
 r  N  r 
 
 is the total number of combinations as selecting i and n-i
 i  n  i 
elements from groups 1 and 2, respectively.
How to obtain the hypergeometric probability distribution:
(a) Using table of Poisson distribution.
(b) Using computer
 by some software, for example, Excel or Minitab.
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Online Exercise:
Exercise 7.3.1
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