Probability

The probabilities of events will always be between 0 and 1 (the probability of the total sample space)

0 \leq P (A) \leq 1

Axioms of Probability

$P (S) = 1$

the total sample space (all events) has probability 1(100%)

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

If $A \cap B = \emptyset$ then the 2 events are mutually exclusive (no elements in common)

For $A_{1}, A_{2}, \dots$ all mutually exclusive, $P (A_{1} \cup A_{2} \cup \dots) = P (A_{1}) + P (A_{2}) + \dots$

$A_{i} \cap A_{j} = \emptyset$

$0 \leq P (A \cap B) \leq m i n (P (A), P (B))$

intersection will always output a smaller set

Conditional probability

Rule of total Probability

If the sample space S can be partitioned into events $B_{1}, B_{2}, \dots B_{k}$ and $P (B_{k}) \neq 0$ $\forall i = 1, 2 \dots k$
Then for any event A in S:

P (A) = P (A | B_{1}) P (B_{1}) + P (A | B_{2}) P (B_{2}) + \dots + P (A | B_{k}) P (B_{k}) = \sum_{i = 1}^{k} P (A | B_{i}) P (B_{i})

or rule of average conditional probability

Probability/Cumulative Discrete Distribution Functions
- PDF
- CDF
- a

Expectation
Variance

Discrete Distributions