CK-12-Basic Probability and Statistics Concepts - A Full Course

http://www.ck12.org Chapter 1. Independent and Dependent Events

If you look at the diagram, you see that the calculation involves not onlyP(A)andP(B), but alsoP(A∩B). However,
the items inA∩Bare also part of eventAand eventB. To represent the probability ofAorB, we need to subtract
theP(A∩B); otherwise, we are double counting. In other words:

P(AorB) =P(A)+P(B)−P(AandB)

or

P(A∪B) =P(A)+P(B)−P(A∩B)

where∩representsandand∪representsor.

This is known as theAddition Principle (Rule).

Addition Principle

P(A∪B) =P(A)+P(B)−P(A∩B)

Think about the idea of rolling a die. Suppose eventAis rolling an odd number with the die, and eventBis rolling a
number greater than 2.

EventA={ 1 , 3 , 5 }

EventB={ 3 , 4 , 5 , 6 }

Notice that the sets containing the possible outcomes of the events have 2 elements in common. Therefore, the
events are mutually inclusive.

Now take a look at the example below to understand the concept of double counting.

Example A

What is the probability of choosing a card from a deck of cards that is a club or a ten?

P(A) =probability of selecting a club

P(A) =

13

52

P(B) =probability of selecting a ten

P(B) =

4

52

P(A∩B) =

1

52

P(A∪B) =P(A)+P(B)−P(A∩B)

P(A∪B) =

13

52

+

4

52

−

1

52

P(A∪B) =

16

52

P(A∪B) =

4

13