Chapter 9: Probability of Combined Events

9.1 Combined Events

• A combined event is an event resulting from the union or intersection of two or more events. (p.244)
• A sample space is the set of all possible outcomes of an experiment. (p.244)
• Number of possible outcomes of two events: n(S) = n(A) × n(B). (p.245)
• When listing outcomes, write them as ordered pairs, e.g. (T, H); with/without replacement changes the sample space. (p.245)

9.2 Dependent and Independent Events

• Independent events: the occurrence of event A has no effect on the occurrence of event B, and vice versa. (p.246)
• Dependent events: the occurrence of event A affects the occurrence of event B. (p.246)
• Without replacement, the first selection changes the probability of the second: dependent events. (p.247)
• With replacement, the second selection has the same probability as the first: independent events. (p.247)
• Probability of event A: P(A) = n(A) / n(S); P(A) = 0 means A definitely does not occur; P(A) = 1 means A definitely occurs. (p.248)
• Multiplication rule (independent events): P(A and B) = P(A ∩ B) = P(A) × P(B). (p.249)
• For three independent events: P(A ∩ B ∩ C) = P(A) × P(B) × P(C). (p.249)
• A tree diagram displays all the possible outcomes; each branch is a possible outcome and probabilities are multiplied along branches. (p.251)
• For dependent events (without replacement), the numerator and denominator of the second probability change after the first event. (p.251)

9.3 Mutually Exclusive and Non-Mutually Exclusive Events

• Mutually exclusive events cannot happen at the same time (no common outcome); A ∩ B = ∅. (p.253)
• Non-mutually exclusive events can happen together: they have at least one common outcome. (p.253)
• Whether two events are mutually exclusive depends on the context (e.g. the group of pupils chosen). (p.254)
• Addition rule, mutually exclusive: P(A ∪ B) = P(A) + P(B), where P(A ∩ B) = 0. (p.256)
• Addition rule, non-mutually exclusive: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). (p.256)
• P(A or B) = P(A ∪ B); P(A and B) = P(A ∩ B). (p.256)
• Check P(A ∩ B) first to decide which form of the addition rule to use. (p.256)
• Venn diagram helps determine P(A ∪ B): add the regions or use the addition rule. (p.257)
• Complement of an event: P(A′) = 1 − P(A). (p.vii)

9.4 Application of Probability of Combined Events

• Expected number of times = probability × number of trials, e.g. P(at least one perfect square) × 540. (p.262)
• “At least one”: add the favourable branches, or use 1 − P(none). (p.262)
• Complement method: P(different colours) = 1 − P(same colours). (p.263)
• Dependent combined events: draw a tree diagram with probabilities that change after each selection; mutually exclusive branches are then added. (p.263)