Probability - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {17}Pr(A) + Pr(B). This justifies the formula given by the addition theorem (property 5). Wecan demonstrate the result more formally, as follows.ProofTo devise a formal proof of the addition theorem, we construct mutually exclusiveevents in a helpful way, and use the third axiom of probability.We can express the union of A and B as A ∪ B = A ∪ (A ′ ∩ B), and the events A andA ′ ∩B are mutually exclusive. Also, B = (A ∩B)∪(A ′ ∩B), and the two events A ∩Band A ′ ∩ B are mutually exclusive. By applying the third axiom to the first of theserelationships, we get thatPr(A ∪ B) = Pr(A) + Pr(A ′ ∩ B).And using the third axiom for the second relationship, we find thatPr(B) = Pr(A ∩ B) + Pr(A ′ ∩ B)=⇒ Pr(A ′ ∩ B) = Pr(B) − Pr(A ∩ B).Hence, Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B), as required.Exercise 4Lego sells ‘minifigures’. There are 16 distinct minifigures, and they are distributed at randomamong shops (it is said). Furthermore, the minifigures cannot be identified prior topurchase and removal of the packaging. A young child wishes to obtain the ‘Pirate Captain’,one of the 16 figures in Series 8. She persuades her parent to buy a minifigure at twodifferent shops. Define the events:• A = “Pirate Captain is purchased at the first shop”• B = “Pirate Captain is purchased at the second shop”.Assume that the distribution of the minifigures is random across shops, and assume thatPr(A ∩ B) = 1256 .abcWhat is Pr(A)? Pr(B)?What is the probability of the child’s wishes being satisfied?Describe each of the following events in words:i B ′ ii A ′ ∪ B ′ iii A ∩ B ′ iv A ′ ∩ B ′ .d Which event is more probable: A ∪ B or A ′ ∩ B ′ ?