Revision sheet

Basics

• Equally likely: favorable ÷ total.
• $P(\text{not } A) = 1 - P(A)$; "at least one" via the complement.

Watch out: make sure the outcomes really are equally likely.

Addition rule

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$; for disjoint events the last term is 0.

Watch out: disjoint is not the same as independent.

Conditional probability

$P(A \mid B) = P(A \text{ and } B) / P(B)$.

Watch out: $P(A \mid B)$ and $P(B \mid A)$ are different.

Independence

• Independent: $P(A \text{ and } B) = P(A)P(B)$.
• In general: $P(A)P(B \mid A)$.

Watch out: without replacement, update the second probability.

Trees

• Multiply along paths; add paths to the same end.
• $P(A \mid B) = P(A \text{ and } B)/P(B)$.

Watch out: with rare events, most positives can be false.

Long run

• Probability = long-run relative frequency.
• Simulate with random numbers.

Watch out: no memory, no "due" outcomes.