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Hypergeometric Distribution

Is a discrete probability distribution, involving a series of dependant trials, with more than one type of success or failure. Involves a number of n draws from a limited population, without replacement. Phrases such as "not replaced" and "not put back" hint at hypergeometric distribution to be required. The sum of xP(x) is expected outcome.

Hypergeometric Trials vs. Bernoulli Trials

Bernoulli trials are independent of one another and all have the same probability, whereas hypergeometric trials are dependant and have varying probabilities. Bernoulli trials have only one type of success and one type of failure, whereas hypergeometric trials have more than one type of success and more than one1 type of failure.

Formula

The formula of a hypergeometric distribution is given by:


where:

x1= # of successful trials of type a
x2= # of successful trials of type b
x3= # of successful trials of type c
a= total possible # of elements in type a trial
b= total possible # of elements in type b trial
c= total possible # of elements in type c trial
n= total # of trials = x1+x2+x3

What is the probability of a Formula 1 race finishing with; 2 Ferrari, 2 Renault, and 1 Honda in the top 5 if each team has 5 cars in the race and the race consists of only those teams?

let x1 = # of successful trials of type a

= # of times Ferrari finishes in the top 5
= 2 Ferrari's

let x2 = # of successful trials of type b

= # of times Renault finishes in the top 5
= 2 Renault's

let x3 = # of successful trials of type c

= # of times Honda finishes in the top 5
= 1 Honda

let a = total possible # of elements in type a trial

= # of Ferrari's in the race
= 5

let b = total possible # of elements in type b trial

= # of Renault's in the race
= 5

let c = total possible # of elements in type c trial

= # of Honda's in the race
= 5

n= total # of trials = x1+x2+x3 = 5

= 0.1665

Therefore, probability of this happening is 0.1665.

There are five bananas and seven oranges in the refrigerator. Four fruits are chosen at random to serve guest. What is the probability that exactly two of the fruits will be oranges?








a=# of Oranges
=7

b=# of Bananas
=5

n=# of trials(# of fruits to be chosen)
=4

x=# of successful trials(# of oranges require)
=2

Expected Values of Hypergeometric Distribution

The expected values of a hypergeometric distribution can be given by:


where:

X= number of successes
s= total # of items in a population which would be seen as a success
f= total # of items in a population which would be seen as a failure
n= # of trials

A jar of jellybeans contains 20 yellow jellybeans and 25 red jellybeans. If 5 jellybeans were drawn from the jar randomly, what is the expected number of red jellybeans drawn?

let n = 5
let s = 25
let f = 20

There are five bananas and seven oranges in the refrigerator. Four fruits are chosen at random to serve guest. What is the expected number of oranges chosen?

a=# of Oranges
=7

b=# of Bananas
=5

n=# of trials(# of fruits to be chosen)
=4





Therefore, you can expect 2.3 oranges.