Addition of probabilities
Mutually exclusive events (also called disjoint events) are events that cannot occur at the same time.
If one event happens, the other event(s) cannot happen in the same experiment.
Mathematically,
P(A∩B) = 0
How to Determine if Events are Mutually Exclusive
Step 1: List the outcomes of each event.
Step 2: Find the intersection (common outcomes).
If there are no common outcomes, the events are mutually exclusive.
If there is at least one common outcome, the events are not mutually exclusive.
Example 1: Rolling a Die
Roll a fair six-sided die.
Sample Space: S = {1, 2, 3, 4, 5, 6}
Event A: Roll a 2 A = {2}
Event B: Roll a 5 B = {5}
Intersection: A∩B = ∅
Since the intersection is empty, P(A∩B) = 0 “The probability of A and B occurring together is zero.”
Answer: Events A and B are mutually exclusive.
Example 2: Rolling a Die
Roll a fair six-sided die.
Sample Space: S = {1, 2, 3, 4, 5, 6}
Event A: Roll an even number A = {2, 4, 6}
Event B: Roll a number greater than 3 B = {4, 5, 6}
Intersection: A∩B = {4, 6}
Since the intersection is empty, P(A∩B) ≠ 0 “The probability of A and B occurring together is greater than zero”
Answer: The events are not mutually exclusive.
Example 3: Choosing a Student
A student is selected from a class.
Sample Space: S = {Male, Female}
Event A: The student is male A = {Male}
Event B: The student is female B = {Female}
Intersection: A∩B = ∅
Since the intersection is empty, P(A∩B) = 0 “The probability of A and B occurring together is zero”
Answer: Events are mutually exclusive.
Example 4: Drawing a Card
A student is selected from a class.
Sample Space: S = {A, 2-10, J, Q, K (spade, hearts, clubs and diamond)}
Event A: Draw a Heart A = {A, 2-10, J, Q, K (hearts)}
Event B: Draw a King B = {King of Hearts}
Intersection: A∩B = {King of Hearts}
Since the intersection is empty, P(A∩B) ≠ 0 “The probability of A and B occurring together is greater than zero”
Answer: Events are not mutually exclusive.
Calculating the Probability of the Complement of an Event and the Union of Two Events
There are two important probability concepts:
Probability of the Complement of an Event
Probability of the Union of Two Events
1. Probability of the Complement of an Event
The complement of an event is the event that does not occur.
If event A occurs, then its complement, denoted by A′ or Ac.
Examples:
If A = getting a Head, then A′ = getting a Tail.
If A = drawing a red card, then A′ = drawing a black card.
Formula:
P(Ac) = 1 − P(A) “Probability of "Not A" = 1 − Probability of A”
Example 1: Rolling a Die
A fair die is rolled. What is the probability of not rolling a 5?
Step 1: Find the probability of rolling a 5.
Total possible outcomes = 6 (1, 2, 3, 4, 5, 6)
Favorable outcomes for rolling a 5 = 1
Prolling a 5 = 1/6
P(Ac) = 1 − P(A)
P(5c) = 1 − 1/6
P(5c) = 5/6
Example 2: Drawing a Card
A card is drawn from a standard deck.
Step 1: Find the probability that the card is not an Ace.
Total possible outcomes = 52
Favorable outcomes not an ACE = 4
P(Ace) = 4/52
P(Ac) = 1 − P(A)
P(Acec) = 1 − 4/52
P(Acec) = 48/52
P(Acec) = 12/13
2. Probability of the Union of Two Events
The union of two events means that either event A, event B, or both events occur.
The symbol is: A∪B read as: “A union B”
P(A∪B) = P(A) + P(B) − P(A∩B) is read as: "The probability of A union B equals the probability of A plus the probability of B minus the probability of A intersection B."
This formula subtracts the overlap so it is not counted twice.
Example 3: Rolling a Die
A die is rolled. Find the probability of getting an even number or a number greater than 4.
Step 1: Get the Event and Intersection
Event A: Even numbers
A = {2, 4, 6}
P(A) = 3/6
Event B: Greater than 4
B = {5, 6}
P(B) = 2/6
Intersection
A∩B = {6}
P(A∩B) = 1/6
Step 2: Substitute to Union Formula
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A∪B) = 3/6 + 2/6 − 1/6
P(A∪B) = 4/6
P(A∪B) = 2/3
Example 4: Drawing a Card
A card is drawn from a deck. Find the probability of drawing a King or a Heart.
Step 1: Get the Event and Intersection
Event A: King
A = {4}
P(K) = 4/52
Event B: Heart
B = {13}
P(B) = 13/52
Intersection
A∩B = {1}
P(A∩B) = 1/52
Step 2: Substitute to Union Formula
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A∪B) = 4/52 + 13/52 − 1/52
P(A∪B) = 16/52
P(A∪B) = 4/13