STATZIKI

Input

How big is the total population? (N)

How many tries are we doing? (n)

How many successful observations is in the total sample? (r)

How many successes do we want to find? (y)

Formulas and Output

\[\frac{\left(\begin{array}{c}r\\ y\end{array}\right) \times \left(\begin{array}{c}N-r\\ n-y\end{array}\right)}{\left(\begin{array}{c}N\\ n\end{array}\right)} = \frac{\left(\begin{array}{c} 5 \\ 3\end{array}\right) \times \left(\begin{array}{c}10-5\\ 3-3\end{array}\right)}{\left(\begin{array}{c}10\\ 3\end{array}\right)} =0.08333\] \[p(y < 3 ) = 1 - p(y \geq 3) = 1 - 0.08333 = 0.91667\] \[p(y \leq 3) = 1 - p(y > 3) = 1 - -0.0 = 1.0\] \[p(y > 3) = 1 - p(y \leq 3) = 1 - 1.0 = -0.0\] \[p(y \geq 3) = 1 - p(y < 3) = 1 - 0.91667 = 0.08333\] \[E(Y) = \frac{nr}{N} = \frac{ 3 \times 5 }{ 10 } = 1.5\] \[V(Y) = n\left(\frac{r}{N}\right)\left(\frac{N-r}{N}\right)\left(\frac{N-n}{N-1}\right) = 3\left(\frac{ 5 }{ 10 }\right)\left(\frac{ 10-5 }{ 10 }\right)\left(\frac{ 10-3 }{ 10-1}\right) = 0.58333\]

Probability Distribution - Hypergeometric

When do you use the hypergeometric distribution?

Use the hypergeometric distribution when the population is small and when the sampling is without replacement.

If n / N < 0.1 it can be approximated using the binomial distribution.

Example, an urn contains ten marbles, of which five are green, two are blue and three are red. Three marbles are to be drawn from the urn,
one at a time without replacement.
- What is the probability that all three marbles drawn will be green?
- What is the probability to get less than 3, more than 3, less than or equal to 3 and more than or equal to 3 green marbles?