# Hypergeometric \defl{Hypergeometric distribution has the following properties:}

1. When units are selected from a finite population without replacement and the population consists of successes and failures.
2. The major difference between the Hypergeometric distribution and the Binomial distribution is that the probability of selecting a success is {\bf not constant and is not independent} from each draw.

\defm{Hypergeometric Distribution}
$P(X=x)=f(x) = \frac{{A\choose x}{{N-A}\choose {n-x}}}{{N\choose n}}$

• $X$ has a Hypergeometric distribution.
• $x$ is the number of successes in the sample
• $n$ is the sample size
• $A$ is the number of successes in the population
• $N$ is the population size
• $N-A$ is the number of failures in the population
• $n-x$ is the number of failures in the sample
• $\mu = E(X) = \frac{nA}{N}$
• $\sigma^2 = V(X) = \left(\frac{N-n}{N-1}\right) \frac{nA(N-A)}{N^{2}}$
• $\frac{N-n}{N-1}$ is called the finite population correction factor.

\defs{Examples}

• The number of spades selected when 5 cards are drawn from a standard 52 deck of cards.
• There are 20 Sony CD players in stock at the Sony store at the Mall Bangkapi; 5 are defective. A customer buys 6 of the 20 CD players. The number of defective CD players bought of the 6 CD players.
• The computer lab has 20 computers and 15 of the 20 computers have illegal software on them. The number of computers selected with illegal software from a sample of size 5.