AP®StatisticsCollege BoardRevision NotesUnit 4: Probability, Random Variables & Probability DistributionsProbabilityMultiplication Rule & Independent Events

Multiplication Rule & Independent Events (College Board AP® Statistics)

Revision Note

Dan Finlay

Expertise

Maths Lead

Multiplication rule

What is the multiplication rule?

  • The multiplication rule is used to find the probability of the intersection of two events

    • i.e. the probability that both events occur

  • To find the probability of Aand B you can:

    • either multiply the probability of A occurring by the probability of B occurring given that A has occurred

    • or multiply the probability of B occurring by the probability of A occurring given that B has occurred

  • P open parentheses A intersection B close parentheses equals P open parentheses A close parentheses times P open parentheses B vertical line A close parentheses or P open parentheses A intersection B close parentheses equals P open parentheses B close parentheses times P open parentheses A vertical line B close parentheses

    • These formulas can be derived by rearranging the conditional probability formula

Exam Tip

Look out for scenarios that involve sampling without replacement. These could involve conditional probabilities.

Worked Example

Chad has 25 comic books, 12 of which involve the superhero Dr Data. Chad chooses two different comic books at random to take on vacation with him.

Calculate the probability that both of the chosen comic books involve Dr Data.

Answer:

Calculate the probability that the first chosen comic book involves Dr Data

There are 25 choices and 12 of them involve Dr Data

P open parentheses first space is space D r space D a t a close parentheses equals 12 over 25

The first comic book is not replaced once chosen, therefore the probability for the second comic book is a conditional probability

Calculate the probability that the second comic book involves Dr Data given that the first one does

If the first chosen comic involves Dr Data then there are 24 comics remaining and 11 of them involve Dr Data

P open parentheses second space is space D r space D a t a vertical line first space is space D r space D a t a close parentheses equals 11 over 24

Find the probability that both comics involve Dr Data by using the multiplication rule

P open parentheses both space are space D r space D a t a close parentheses equals P open parentheses first space is space D r space D a t a close parentheses times P open parentheses second space is space D r space D a t a vertical line first space is space D r space D a t a close parentheses

table row cell P open parentheses both space are space D r space D a t a close parentheses end cell equals cell 12 over 25 times 11 over 24 end cell row blank equals cell 11 over 50 end cell end table

The probability that both of the chosen comic books involve Dr Data is 11 over 50 (or 0.22)

Independent events

What are independent events?

  • Independent events are events that are not affected by the occurrence of each other

    • The probability of an event occurring does not change if the other event has occurred

      • e.g. rolling a six on a dice and a coin landing on tails are independent events

      • e.g. rolling a six and rolling an even number on the same dice roll are not independent events

How can I check whether two events are independent?

  • If A and B are independent events then P open parentheses A vertical line B close parentheses equals P open parentheses A close parentheses and P open parentheses B vertical line A close parentheses equals P open parentheses B close parentheses

  • The multiplication rule for independent events simplifies to P open parentheses A intersection B close parentheses equals P open parentheses A close parentheses times P open parentheses B close parentheses

    • This can be extended to more than two events

      • e.g. if A, B and C are independent events then P open parentheses A intersection B intersection C close parentheses equals P open parentheses A close parentheses times P open parentheses B close parentheses times P open parentheses C close parentheses

  • To check whether A and B are independent, check if one of the following equivalent statements is true:

    • P open parentheses A vertical line B close parentheses equals P open parentheses A close parentheses

    • P open parentheses B vertical line A close parentheses equals P open parentheses B close parentheses

    • P open parentheses A intersection B close parentheses equals P open parentheses A close parentheses times P open parentheses B close parentheses

Exam Tip

Do not assume two events are independent unless you are told in the question!

Worked Example

A college lecturer surveys a large group of first-year students about their accommodation. Students are asked whether they live on campus or live elsewhere. They are also asked whether they have a private or shared bathroom. The relative frequency of each category is shown in the table.

Private bathroom

Shared bathroom

Total

Campus

0.232

0.416

0.648

Elsewhere

0.098

0.254

0.352

Total

0.330

0.670

1.000

For the students surveyed, are the events of living on campus and having a private bathroom independent? Justify your answer.

Answer:

Identify the relevant probabilities

table row cell P open parentheses campus close parentheses end cell equals cell 0.648 end cell row cell P open parentheses private close parentheses end cell equals cell 0.330 end cell row cell P open parentheses campus space and space private close parentheses end cell equals cell 0.232 end cell end table

There are three methods to check whether two events are independent

Method 1: Checking P open parentheses A vertical line B close parentheses equals P open parentheses A close parentheses

P open parentheses campus vertical line private close parentheses equals P open parentheses campus close parentheses

Calculate the probability that a student lives on campus given that they have a private bathroom

Use the formula P open parentheses campus vertical line private close parentheses equals fraction numerator P open parentheses campus space and space private close parentheses over denominator P open parentheses private close parentheses end fraction

table row cell P open parentheses campus vertical line private close parentheses end cell equals cell fraction numerator 0.232 over denominator 0.330 end fraction end cell row blank equals cell 0.703... end cell end table

Compare this to the probability that a student lives on campus

0.703... not equal to 0.648

The events of living on campus and having a private bathroom are not independent because P open parentheses campus vertical line private close parentheses not equal to P open parentheses campus close parentheses

Method 2: Checking P open parentheses B vertical line A close parentheses equals P open parentheses B close parentheses

P open parentheses private vertical line campus close parentheses equals P open parentheses private close parentheses

Calculate the probability that a student has a private bathroom given that they live on campus

Use the formula P open parentheses private vertical line campus close parentheses equals fraction numerator P open parentheses private space and space campus close parentheses over denominator P open parentheses campus close parentheses end fraction

table row cell P open parentheses private vertical line campus close parentheses end cell equals cell fraction numerator 0.232 over denominator 0.648 end fraction end cell row blank equals cell 0.358... end cell end table

Compare this to the probability that a student has a private bathroom

0.358... not equal to 0.330

The events of living on campus and having a private bathroom are not independent because P open parentheses private vertical line campus close parentheses not equal to P open parentheses private close parentheses

Method 3: Checking P open parentheses A intersection B close parentheses equals P open parentheses A close parentheses times P open parentheses B close parentheses

P open parentheses campus space and space private close parentheses equals P open parentheses campus close parentheses times P open parentheses private close parentheses

Multiply together the probability that a student lives on campus and the probability that a student has a private bathroom

table row cell P open parentheses campus close parentheses times P open parentheses private close parentheses end cell equals cell 0.648 times 0.330 end cell row blank equals cell 0.21384 end cell end table

Compare this to the probability that a student lives on campus and has a private bathroom

0.21384 not equal to 0.232

The events of living on campus and having a private bathroom are not independent because P open parentheses campus space and space private close parentheses not equal to P open parentheses campus close parentheses times P open parentheses private close parentheses

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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.