1. Complementary and mutually exclusive events
Probability 12B
Complementary and mutually exclusive events

2. 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′)

3. 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

4. Worked example
In a deck of cards, there are 4 suits (diamonds, hearts, spades, clubs)
That means there are cards per suit, or 13 of each suit
Pr(spade) = = = 0.25
Pr(spadeꞌ) = 1 – Pr(spade)
= 1 – 0.25
= 0.75

5. 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 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

6. 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)

7. Worked example
A card is drawn from a pack of 52 playing cards. What is the probability that the card is a heart or a club?
Steps:
Are the events (heart or club) mutually exclusive?
Yes because they have no common/overlapping elements
Work out the probability of getting a heart or a club
Pr(heart) = = = 0.25
Pr(club) = = = 0.25
Write the addition law for mutually exclusive events
Pr(A ∪ B) = Pr(A) + Pr(B) Let A = heart; B = club
Pr(A ∪ B) = = 0.5
Note: you can also use the theoretical probability formula
Pr(heart or club) = = = 0.5

8. WORKED EXAMPLE
Pr(an odd number) = = 1 2
Pr(less than 4) = = 1 2
Pr(an odd number or a number less than 4) are NOT mutually exclusive, as they share common elements {1, 3}
Pr(odd AND <4) = = 1 3
Write the additional law for non-mutually exclusive events:
Pr(A ∪ B) = Pr(A) + Pr(B) - Pr(A ∩ B)
Pr(odd ∪ <4) = Pr(odd) + Pr(<4) – Pr(odd ∩ <4) =
= 2 3
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, 5}

9. Questions to do
Exercise 12B p400: 1, 3, 6, 10, 13, 14, 15, 20