Probability
Probability
This domain covers conditional probability, independence, total probability, Bayes’ theorem, and random variables. For events A and B with P(B)>0, P(A|B)=P(A∩B)/P(B). Rearranging gives the multiplication rule P(A∩B)=P(B)P(A|B). Independence is a special condition, not a synonym for mutually exclusive.
Core rules
- Mutually exclusive: P(A∩B)=0. Independent: P(A∩B)=P(A)P(B).
- Total probability: if B1, B2, … partition the sample space, P(A)=ΣP(Bi)P(A|Bi).
- Bayes’ theorem reverses a conditional probability: P(Bi|A)=P(Bi)P(A|Bi)/P(A).
- For a random variable X, mean E(X)=ΣxP(x); variance Var(X)=E(X^2)−[E(X)]^2.
- Binomial probability: P(X=r)=C(n,r)p^r(1−p)^{n−r}.
Worked example
A bag has 3 red and 2 blue balls. Two balls are drawn without replacement. Let A be first red and B be second red. P(A)=3/5. Given A, 2 red remain among 4 balls, so P(B|A)=2/4=1/2. Therefore P(A∩B)=P(A)P(B|A)=(3/5)(1/2)=3/10. With replacement, the conditional probability would remain 3/5 and the product would be 9/25.
Traps
Using independence when draws are without replacement; confusing P(A|B) with P(B|A); using total probability without a complete partition; forgetting that mutually exclusive events are not independent unless one has probability zero; omitting the binomial coefficient.