Probability

10 Feb 2018 on Math

1. frequentist probability

- Objectivists assign numbers to describe some objective or physical state of affairs.

2. Bayesian probability

- Subjectivists assign numbers per subjective probability.

3. Independent Events

$P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B)$

4. Mutually exclusive events

$P(A{\mbox{ and }}B)=P(A\cap B)=0$

5. summary of probability

- A

$P(A)\in [0,1]$

- not A

$P(A^{\complement })=1-P(A)\,$

- A or B

${\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}$

- A and B

${\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}$

- A given B

$P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,$

6. Kolmogorov axioms

- first axiom

$P(A)\in\mathbb{R}, P(A)\geq 0$

- second axiom

$P(\Omega) = 1$

- third axiom

$A \cap B = \emptyset \;\;\; \rightarrow \;\;\; P(A \cup B) = P(A) + P(B)$

Reference

Probability - wikipedia

Probability Axioms- wikipedia