Probability - the Australian Mathematical Sciences Institute

A guide for teachers – Years 11 and 12 • {15}This property has a useful application. We write A ⇒ B to mean that, if A occurs, then Boccurs. So A ⇒ B is the same as A ⊆ B, which implies Pr(A) ≤ Pr(B). For example:• Indiana Jones can only find the treasure (A)• if he first solves the puzzle of the seven serpents (B);so A ⊆ B and therefore Pr(A) ≤ Pr(B); his chance of finding the treasure is at most equalto his chance of solving the puzzle.While property 1 is important, it is rather obvious.Property 20 ≤ Pr(A) ≤ 1, for each event A.ProofThe first axiom says that Pr(A) ≥ 0. It is true for every event A that A ⊆ E . HencePr(A) ≤ Pr(E ), by property 1. But Pr(E ) = 1, by the second axiom, and so it followsthat Pr(A) ≤ 1.This property formalises the scale for probabilities, given the axioms.û û û û ûûû û ûû ûû Øû ûû ûûû û û û ûûû û ûûûûûûεThe empty event.Property 3Pr(∅) = 0.ProofChoose any event A. Then the two events A and the empty set ∅ are mutuallyexclusive, so Pr(A ∪ ∅) = Pr(A) + Pr(∅), by the third axiom. Since A ∪ ∅ = A, thisgives Pr(A) = Pr(A) + Pr(∅). Hence, Pr(∅) = 0.It would be rather strange if the probability of the empty set was anything other thanzero, so it is reassuring to confirm that this is not so: Pr(∅) = 0, as expected.