Probability
Combining probabilities
- Addition ("or"): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
- Multiplication ("and", if independent): $P(A \cap B) = P(A)\,P(B)$.
- Mutually exclusive events can't both happen; independent events don't affect each other.
Practice
P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.2. What is P(A∪B)?
P(A∪B) = 0.5 + 0.4 − 0.2 = 0.7.
Conditional probability
- Conditional: the chance of $A$ given $B$ has happened:
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$
- Independence test: check whether $P(A \cap B) = P(A)\times P(B)$.
Practice
P(A) = 0.5, P(B) = 0.4 and P(A∩B) = 0.2. Are A and B independent?
P(A)×P(B) = 0.5 × 0.4 = 0.2, which equals P(A∩B), so they are independent.
Practice
P(A∩B) = 0.2 and P(B) = 0.4. What is P(A | B)?
P(A|B) = P(A∩B)/P(B) = 0.2/0.4 = 0.5.
You've got it
Key idea
- addition (or): $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
- multiplication (and, independent): $P(A\cap B) = P(A)P(B)$
- conditional: $P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}$; test independence with $P(A\cap B) = P(A)P(B)$