Probability
Probability tests counting, the rules of probability, conditional probability, independence, Bernoulli trials, and expected value. Higher Level questions often place these ideas in a context that requires careful interpretation of sampling with or without replacement.
Core rules
- Addition rule: (P(A\cup B)=P(A)+P(B)-P(A\cap B)). Mutually exclusive events have (P(A\cap B)=0).
- Conditional probability: (P(A\mid B)=\frac{P(A\cap B)}{P(B)}). Events are independent when (P(A\cap B)=P(A)P(B)).
- Counting: permutations ({}^{n}P_r=\frac{n!}{(n-r)!}); combinations ({}^{n}C_r=\frac{n!}{r!(n-r)!}). Use combinations when order does not matter.
- Bernoulli trials: (P(r\text{ successes in }n\text{ trials})={}^{n}C_r p^r(1-p)^{n-r}).
- Expected value: (E(X)=\sum xP(x)).
Worked example
A bag contains 5 red and 3 blue counters. Two are drawn without replacement. Find the probability that the second is red given that the first is blue. After a blue counter is removed, 7 counters remain, of which 5 are red. Therefore (P(\text{second red}\mid\text{first blue})=\frac57).
Traps
Treating without-replacement draws as independent; confusing (P(A\mid B)) with (P(A\cap B)); assuming mutually exclusive events are independent; using permutations when order is irrelevant; forgetting the complement for “at least one”; failing to adjust the denominator after a draw.