PROBABILITY

The average of the number
$\bar{x}\ =\ \frac{\sum x}{n}$

The DIFFERENCE between the

MAXIMUM value
to the
MINIMUM value
${\rm Value}_{max}\ -\ {\rm Value}_{min}$

Population Standard Deviation
$\sigma_p=\ \sqrt{\frac{\sum\left(x-\bar{x}\right)^2}{n}}$

Sample Standard Deviation
$\sigma_s=\ \sqrt{\frac{\sum\left(x-\bar{x}\right)^2}{n-1}}$

$Q_i=\ \frac{i}{4}\left(n\ +\ 1\right)\ $

IMPORTANT = PERMUTATION

NOT IMPORTANT = COMBINATION

$n\ P\ r\ =\ \frac{n!}{\left(n\ -r\right)!}$

$n\ P\ r\ =\ \frac{N!}{R!}$
N= number of obj. (counting certain identical obj as 1)
R= number of how many times the certain obj repeated

$n\ P\ r\ =\ \left(n-1\right)!$

$n\ P\ r\ =\ \frac{\left(n-1\right)!}{2}$

$n\ C\ r\ =\ \frac{n!}{r!\ \left(n-r\right)!}$
n= number of choices
r= number of outcomes

$!n\ =n!\sum_{k=0}^{n}{\ \frac{\left(-1\right)^k}{k!}}$

$n\ C\ r=\ \frac{n\ P\ r}{r\ !}$

$P\ =\ \frac{SUCCESSFUL\ outcomes}{total\ number\ of\ POSSIBLE\ outcomes}$

$E = pS$
p= probability of success
S= sum of prize

$Odds\ in\ Favor\ =\ \frac{SUCCESSFUL\ }{failure}$

$Odds\ Against\ =\ \frac{failure\ }{SUCCESS}$

$P\left(A\ \cap B\ \right)=\ 0$
$P\left(A\ \cup B\ \right)=\ P\ \left(A\right)\ +P\ \left(B\right)$
the probability of both events happening at the same time is zero.
For example, when tossing a coin, the result can either be heads or tails, but not both

$P\left(A\ \cap B\ \right)\neq\ 0$
$P\left(A\ \cup B\ \right)=\ P\ \left(A\right)\ +P\ \left(B\right)-P\left(A\ \cap B\ \right)$
For example, the events “buying an alarm system” and “buying bucket seats”
are mutually inclusive, as both events can happen at the same time

$P\left(A|B\right) \cdot P\ \left(B\right)=\ P\left(B|A\right) \cdot P\ \left(A\right)\ =P\left(A\cap B\right)$

$ P\ \left(x\ =r\right)=\ \left(n\ C\ r\right)\cdot\left(p^r\right)\cdot{\ \left(q\right)}^{n-r}$

n =number of trials (occurrences)
r =number of successful trials
p =probability of success in a single trial
q = (1-p)

$\bar{x}\ =\ np$
$\sigma\ =\ \sqrt{npq}$