Mathematics · Ch 14 — Probability

Summary

Random experiment & sample space: A random experiment has multiple possible outcomes; the set of all outcomes is the sample space $S$ . An event is any subset of $S$ .
Types of events: Impossible event ( $\emptyset$ ), sure event ( $S$ ), simple event (exactly one outcome), compound event (more than one outcome).
Algebra of events: For events $A$ and $B$ : complement $A'$ or $\bar{A}$ , union $A \cup B$ , intersection $A \cap B$ , difference $A - B = A \cap B'$ . Events are mutually exclusive if $A \cap B = \emptyset$ ; exhaustive if $A \cup B = S$ .
Axiomatic probability: For each event $A$ , $P(A) \ge 0$ ; $P(S) = 1$ ; for mutually exclusive events $A_1, A_2, \dots$ , $P(\bigcup A_i) = \sum P(A_i)$ .
Key formulas:
- $0 \le P(A) \le 1$
- $P(\emptyset) = 0$
- $P(A') = 1 - P(A)$
- $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Equally likely outcomes: If $S$ has $n$ equally likely outcomes, $P(A) = \frac{\text{number of outcomes in } A}{n}$ .
Conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ , provided $P(B) \neq 0$ .
Multiplication theorem: $P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$ .
Independent events: $A$ and $B$ are independent iff $P(A \cap B) = P(A) \cdot P(B)$ . Independence implies $P(A|B) = P(A)$ and $P(B|A) = P(B)$ .
Partition & total probability: If events $E_1, E_2, \dots, E_n$ partition $S$ (mutually exclusive and exhaustive), then for any event $A$ : $P(A) = \sum_{i=1}^n P(E_i) \cdot P(A|E_i)$ .
Bayes' theorem: For the same partition, $P(E_k|A) = \frac{P(E_k) \cdot P(A|E_k)}{\sum_{i=1}^n P(E_i) \cdot P(A|E_i)}$ .
Random variable & probability distribution: A random variable $X$ assigns a real number to each outcome. Its probability distribution lists $P(X = x_i)$ for each value $x_i$ , with $\sum P(X = x_i) = 1$ .
Mean & variance of a random variable:
- Mean: $\mu = E(X) = \sum x_i \cdot P(X = x_i)$ …

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