Addition Rule of Probability

1. For two events and , the probability that either or occurs is:
2. If and are mutually exclusive events, meaning , then:

 

Example 1. A random number is chosen from the first natural numbers. Find the probability that the number selected is a multiple of either or . Write your answer as a fraction or whole number.

 Solution

Let be the event of choosing a multiple of and be the event of choosing a multiple of , then favorable outcomes is an event of choosing a multiple of .

Since and ,

 

Example 2. A number is selected at random from the first natural numbers. What is the probability that it is less than or greater than ? Write your answer as a fraction or whole number.

 Solution

Let be the event of choosing a number less than and be the event of choosing a number greater than ,
and .

 

Probability of the Complement

For any event in a sample space , and its complement :

The probability of events like “at least”, “greater than or equal to”, or “less than or equal to” can often be calculated easily by using the complement probability.

 

Example. A box contains cards numbered from to . Two cards are drawn at random from the box. Find the probability that the product of the two numbers on the drawn cards is even if both the cards are drawn together.

 Solution

The complementary event of ‘an event in which the product of two numbers is even’ is ‘an event in which the product of two numbers is odd’. Since the probability that a product of two numbers is odd is the probability that both numbers are odd,

Therefore, the probability that the product of two numbers is even is

 

Comparison of Mutually Exclusive Events and Complements

	Mutually Exclusive Events	Complement
Definition	Two events are mutually exclusive if	The complement of event is
Venn Diagram
Meaning	Both events cannot happen at the same time	Event does not occur
Probability