PROBABILITY 12 B COMPLEMENTARY AND MUTUALLY EXCLUSIVE EVENTS

COMPLEMENTARY EVENTS • If we call the outcome we want A, then the opposite of this is written as Aꞌ • This is known as the complement of A. The complement can be referred as ‘not A’ • If A and A′ are complementary events then: • P(A) + P(A′) = 1 • This may be rearranged to P(A′) = 1 − P(A) or P(A) = 1 − P(A′)

EXAMPLE • If we look at events M and Mꞌ, • M is the area in the M circle • M ꞌ is everything that is NOT in the M circle (including the rectangle) • Together, they cover the entire Venn diagram • Pr(M) + Pr(Mꞌ) = 1

WORKED EXAMPLE •

MUTUALLY EXCLUSIVE EVENTS • Two events that have no common elements and that cannot occur simultaneously (together at the same time) are defined as mutually exclusive events. That is A ∩ B = { } or φ (null – no common elements) • Complementary events are mutually exclusive • In a Venn diagram, this means they don’t overlap • If two events A and B are mutually exclusive then Pr(A ∩ B) = 0

ADDITION LAW OF PROBABILITY • If events A and B are NOT mutually exclusive, the Addition Law of probability states that: Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B) • If events A and B ARE mutually exclusive, the Addition Law of probability states that: Pr(A ∪ B) = Pr(A) + Pr(B)

WORKED EXAMPLE •

WORKED EXAMPLE A die is rolled. Set of possibilities = {1, 2, 3, 4, 5, 6} Determine: (a) Pr(an odd number) {1, 3, 5} (b) Pr(a number less than 4) {1, 2, 3} (c) Pr(an odd number or a number less than 4) {1, 2, 3,

QUESTIONS TO DO • Exercise 12 B p 400: 1, 3, 6, 10, 13, 14, 15, 20