Formulas: Probability

Module: M04

Odds

$\text{Odds for }E=\frac{P(E)}{1-P(E)}\text{Odds against }E=\frac{1-P(E)}{P(E)}$

Conditional Probability

$P(A|B)=\frac{P(AB)}{P(B)}$

Addition Rule

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

If mutually exclusive: $P(A\cup B)=P(A)+P(B)$

Multiplication Rule

Dependent: $P(AB)=P(A|B)\cdot P(B)$

Independent: $P(AB)=P(A)\cdot P(B)$

Total Probability Rule

$P(A)={\sum }_{i=1}^{N}P(A|B_i)\cdot P(B_i)$

Bayes’ Formula

$P(B|A)=\frac{P(A|B)\cdot P(B)}{P(A)}$

Expected Value

$E(X)={\sum }_{i=1}^{n}P(X_i)\cdot X_i$

Variance of a Random Variable

${\sigma }^2(X)={\sum }_{i=1}^{n}P(X_i)\cdot [X_i-E(X){]}^2$

Standard Deviation

$\sigma (X)=\sqrt{{\sigma }^2(X)}$

Factorial

$n!=n\times (n-1)\times \cdots \times 2\times 1$

Permutation

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

Combination

$C(n,r)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$

Binomial Probability

$P(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$

where $n$ = trials, $x$ = successes, $p$ = probability of success