AP®StatisticsCollege BoardRevision NotesUnit 4: Probability, Random Variables & Probability DistributionsProbabilityAddition Rule & Mutually Exclusive Events

Addition Rule & Mutually Exclusive Events (College Board AP® Statistics)

Revision Note

Dan Finlay

Expertise

Maths Lead

Addition rule

What is the addition rule?

  • The addition rule is used to find the probability of the union of two events

  • To find the probability of A or B or both

    • Add together the probability of Aand the probability of B

    • Subtract the probability of A and B both occurring

      • This is because it has been counted twice (once for Aand once for B)

  • P open parentheses A union B close parentheses equals P open parentheses A close parentheses plus P open parentheses B close parentheses minus P open parentheses A intersection B close parentheses

    • This formula is given in the exam

  • The addition rule can be used to find the probability of the intersection of two events

    • P open parentheses A intersection B close parentheses equals P open parentheses A close parentheses plus P open parentheses B close parentheses minus P open parentheses A union B close parentheses

Worked Example

The owner of a company buys each employee a sandwich for lunch every day. One day, there are two options: peanut butter and jelly sandwiches or grilled cheese sandwiches.

64% of the employees like peanut butter and jelly sandwiches.

72% of the employees like grilled cheese sandwiches.

5% of the employees do not like either of the options.

Find the probability that a randomly selected employee likes both of the options.

Answer:

Let J be the event that the employee likes peanut butter and jelly sandwiches

Let C be the event that the employee likes grilled cheese sandwiches

Identify the probabilities

64% like peanut butter and jelly sandwiches and 72% like grilled cheese sandwiches

table row cell P open parentheses J close parentheses end cell equals cell 0.64 end cell row cell P open parentheses C close parentheses end cell equals cell 0.72 end cell end table

5% of employees do not like either of the options

table row cell P open parentheses J apostrophe intersection C apostrophe close parentheses end cell equals cell 0.05 end cell end table

This means that 95% of employees like at least one of the options

P open parentheses J union C close parentheses equals 0.95

Use the addition rule to find the probability that the employee likes both of the options

table row cell P open parentheses J union C close parentheses end cell equals cell P open parentheses J close parentheses plus P open parentheses C close parentheses minus P open parentheses J intersection C close parentheses end cell row cell 0.95 end cell equals cell 0.64 plus 0.72 minus P open parentheses J intersection C close parentheses end cell row cell P open parentheses J intersection C close parentheses end cell equals cell 0.41 end cell end table

The probability that a randomly selected employee likes both of the options is 0.41

Mutually exclusive events

What are mutually exclusive events?

  • Mutually exclusive events are events that contain no common outcomes

    • The events cannot happen at the same time

      • e.g. landing on a 5 and landing on an even number on the same dice roll are mutually exclusive events

      • e.g. landing on a 5 and landing on an odd number on the same dice roll are not mutually exclusive events

    • Mutually exclusive events are also called disjoint events

  • Any event and its complement are mutually exclusive

A table with rows labeled Bronze, Silver, and Gold, and columns numbered 1 to 6. Events A (getting gold) and B (getting silver) are highlighted as being mutually exclusive events.
Example of two mutually exclusive events

How can I check whether two events are mutually exclusive?

  • If Aand B are mutually exclusive events then P open parentheses A intersection B close parentheses equals 0

  • The addition rule for mutually exclusive events simplifies to P open parentheses A union B close parentheses equals P open parentheses A close parentheses plus P open parentheses B close parentheses

    • This can be extended to more than two events

      • e.g. if A, B and C are mutually exclusive events then P open parentheses A union B union C close parentheses equals P open parentheses A close parentheses plus P open parentheses B close parentheses plus P open parentheses C close parentheses

  • Too check whether A and B are mutually exclusive, check if one of the following equivalent statements is true:

    • P open parentheses A intersection B close parentheses equals 0

    • P open parentheses A union B close parentheses equals P open parentheses A close parentheses plus P open parentheses B close parentheses

How can I find probabilities involving mutually exclusive events?

  • If Aand B are mutually exclusive events then for any event E

    • P open parentheses E close parentheses equals P open parentheses E intersection A close parentheses plus P open parentheses E intersection B close parentheses

      • This is called the partition theorem

    • This formula is useful in exam questions especially when using a tree diagram or two-way tables

      • e.g. the probability of picking a green marble is equal to:

        • the probability of picking a green marble and a coin landing on heads

        • plus the probability of picking a green marble and a coin landing on tails

Probability table of outcomes with three rows of colors (green, blue, yellow) and two columns of the outcome of throwing a coin (heads and tails). It illustrates mutually exclusive events with labeled equations and sums.
Example of mutually exclusive events in a two-way table

Worked Example

In a high school, 15% of all the students wear glasses and 25% of all the students can write with their left hand. 35% of all the students wear glasses or can write with their left hand. A student is selected at random.

(a) Determine whether the events "the student wears glasses" and "the student can write with their left hand" are mutually exclusive. Explain your answer.

Answer:

Let G be the event that the student wears glasses

Let L be the event that the student can write with their left hand

Identify the probabilities

The 35% of students who wear glasses or can write with their left hand includes the students (if any) who wear glasses and can write with their left hand

table row cell P open parentheses G close parentheses end cell equals cell 0.15 end cell row cell P open parentheses L close parentheses end cell equals cell 0.25 end cell row cell P open parentheses G union L close parentheses end cell equals cell 0.35 end cell end table

If they are mutually exclusive then P open parentheses G union L close parentheses equals P open parentheses G close parentheses plus P open parentheses L close parentheses

Check to see if this formula works for these events

P open parentheses G close parentheses plus P open parentheses L close parentheses equals 0.4 P open parentheses G close parentheses plus P open parentheses L close parentheses not equal to P open parentheses G union L close parentheses

The events "the student wears glasses" and "the student can write with their left hand" are not mutually exclusive because P stretchy left parenthesis G union L stretchy right parenthesis not equal to P stretchy left parenthesis G stretchy right parenthesis plus P stretchy left parenthesis L stretchy right parenthesis

(b) Find the probability that the student wears glasses and cannot write with their left hand.

Answer:

The probability that is needed is P open parentheses G intersection L apostrophe close parentheses

Use the addition rule to find the probability that the student wears glasses and can write with their left hand

table row cell P open parentheses G union L close parentheses end cell equals cell P open parentheses G close parentheses plus P open parentheses L close parentheses minus P open parentheses G intersection L close parentheses end cell row cell 0.35 end cell equals cell 0.15 plus 0.25 minus P open parentheses G intersection L close parentheses end cell row cell P open parentheses G intersection L close parentheses end cell equals cell 0.05 end cell end table

A student can either write with their left hand or they cannot write with their left hand

Therefore the events L and L apostrophe are mutually exclusive so use the partition theorem P open parentheses G close parentheses equals P open parentheses G intersection L close parentheses plus P open parentheses G intersection L apostrophe close parentheses

table row cell 0.15 end cell equals cell 0.05 plus P open parentheses G intersection L apostrophe close parentheses end cell row cell P open parentheses G intersection L apostrophe close parentheses end cell equals cell 0.1 end cell end table

The probability that a student wears glasses and cannot write with their left hand is 0.1

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    Dan Finlay

    Author: Dan Finlay

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.