Probability
Probability tests conditional probability, independence, multiplication theorem, total probability, Bayes’ theorem, random variables, and binomial distribution. The exam often uses tree diagrams or two-stage experiments.
Core rules
P(A|B)=P(A∩B)/P(B), P(B)>0. The multiplication rule is P(A∩B)=P(A)P(B|A). Events are independent if P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A). Total probability: for a partition B₁,B₂,…, P(A)=ΣP(Bᵢ)P(A|Bᵢ). Bayes’ theorem reverses the conditioning: P(Bᵢ|A)=P(Bᵢ)P(A|Bᵢ)/ΣP(Bⱼ)P(A|Bⱼ). For a random variable X, mean=ΣxP(x), variance=Σx²P(x)−(mean)². In binomial trials, P(X=r)=nCr pʳqⁿ⁻ʳ, with q=1−p.
Worked example
Box I has 2 white, 3 black; Box II has 4 white, 1 black. Choose a box at random and draw one ball; it is white. Find P(Box I|white). P(I)=1/2, P(W|I)=2/5; P(II)=1/2, P(W|II)=4/5. Total P(W)=1/2·2/5+1/2·4/5=1/5+2/5=3/5. Bayes gives P(I|W)=(1/2·2/5)/(3/5)=(1/5)/(3/5)=1/3.
Traps
Do not add probabilities of overlapping events without subtracting the intersection. Independent is not the same as mutually exclusive. In without-replacement problems, the denominator changes. In Bayes’ theorem, the denominator must be the total probability of the observed event, not just one branch. Check nCr and the powers of p and q.