A Discrete Distribution with Dependent Bernoulli Trials.
Commonly used when items are removed from the set.
x \geq max \{ 0, n-(N-M) \}\\
x \leq min \{ n, M\}\\
\end{cases}$$
# Formula
The probability of outcomes *x* successful outcomes in *r* trials is:
$$\frac{_{a}C_{x} * _{n-a}C_{r-x}}{_{n}C_{r}}$$
r : # of selections
a : what random variable x will take from. the success set. number of possible successes
n : population total set(universal set fit within the selection space)
### Memorizing the formula
![[Hypergeometric Distribution-20240122210727030.webp]]
![[Pasted image 20240122210828.png]]
# Expectations
$$E(x) = \frac{ra}{n}$$
# Example
Probability distribution of the number of hearts in a 5 card hand from a standard deck of cards
| Number of hearts, x | P(x) | x P(x) |
| ------------------- | ------------------------------------------------------- | ------ |
| 0 | ($\frac{_{13}C_{0} * _{39}C_{5}}{_{52}C_{5}}$) = 0.2215 | 0 |
| 1 | ($\frac{_{13}C_{1} * _{39}C_{4}}{_{52}C_{5}}$) = 0.4114 | 0.4114 |
| 2 | ($\frac{_{13}C_{2} * _{39}C_{3}}{_{52}C_{5}}$) = 0.2743 | 0.5486 |
| 3 | = 0.815 | 0.2445 |
| 4 | = 0.0143 | 0.0572 |
| 5 | = 0.0005 | 0.0025 |
The expected value is the sum of column x P(x). that is 1.26
**What this means is that we are expected to get 1.26 hearts for every 5 card hand**
# Example 2. Find population
in 2002, 300 tagged bass released. in 2003, 200 bass caught and 20 found were tagged. what is the estimate size of bass population in 2003?
n = ?
a = 300
r = 200
E(x) = ra/n
n = ra
n = 3000