Probability

Basic Probability

$0 \leq P(A) \leq 1$ $P(A^c) = 1 - P(A)$

Addition Rule

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

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

Multiplication Rule

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

If $A$ and $B$ are independent: $P(A \cap B) = P(A) \cdot P(B)$.

Conditional Probability

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

Independence

Events $A$ and $B$ are independent if $P(B|A) = P(B)$.

Law of Large Numbers

As the number of trials increases, the sample proportion approaches the true probability.

Two-Way Tables

Use two-way tables to find joint, marginal, and conditional probabilities.

Tree Diagrams

Useful for visualizing sequential events and computing conditional probabilities (Bayes' theorem).

Bayes' Theorem

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

Problem: In a class, 60% are female. Of females, 80% pass. Of males, 70% pass. Find $P(\text{pass})$.

Solution: By the law of total probability: $P(\text{pass}) = P(F)P(\text{pass}|F) + P(M)P(\text{pass}|M)$ $= 0.6(0.8) + 0.4(0.7) = 0.48 + 0.28 = 0.76$

• Addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Multiplication rule: $P(A \cap B) = P(A)P(B|A)$.
• Independence: $P(B|A) = P(B)$, equivalently $P(A \cap B) = P(A)P(B)$.
• Bayes' theorem reverses conditional probabilities.