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Applications of Probability

Constructing and Interpreting Two-Way Frequency Tables

In probability, tables are used to display a data set. For example, frequency tables show how often an outcome appears in a category. To represent a data set that includes two categories, another type of table is needed. This lesson will discuss how to create and interpret these tables.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Chocolate Bar or Fruit?

Zosia attends North High School in Honolulu. She asked students whether they prefer a chocolate bar or a piece of fruit as a lunchtime snack and whether or not they surf. She obtained the following information.

Results of Zosia's Survey
Letting be the event that a student surfs and be the event that a student prefers fruit as a lunchtime snack, Zosia wants to calculate the following probabilities.

Discussion

Two-Way Frequency Tables, Joint Frequency, and Marginal Frequency

A two-way frequency table, also known as a two-way table, is a table that displays categorical data that can be grouped into two categories. One of the categories is represented by the rows of the table, the other by the columns. For example, the table below shows the results of a survey in which participants were asked if they have a driver's license and if they own a car.

two-way table

Here, the two categories are car and driver's license, both with possible answers of yes and no. The entries in the table are called joint frequencies. Two-way frequency tables often include the total of the rows and columns. These totals are called marginal frequencies.

two-way table
The sum of the Total row and the Total column is equal to the sum of all joint frequencies and is called the grand total. In the case of the survey, the grand total is From the table it can be read that, among other things, people both have a driver's license and own a car. It can also be read that people do not have a driver's license.

Discussion

Making a Two-Way Frequency Table

Organizing data in a two-way frequency table can help with visualization, which in turn makes it easier to analyze and present the data. To draw a two-way frequency table, three steps must be followed.

  1. Determine the Categories
  2. Fill the Table With Given Data
  3. Find Any Missing Frequencies

Suppose that people took part in an online survey, where they were asked whether they prefer top hats or berets. Out of the males that participated, of them prefer berets. Also, of the females chose top hats as their preference. The steps listed above will be developed for this example.

1

Determine the Categories

First, the two categories of the table must be determined, after which the table can be drawn without frequencies. Here, the participants gave their hat preference and their gender, which are the two categories. Hat preference can be further divided into top hat and beret, and gender into female and male.

two-way table

The total row and total column are included to write the marginal frequencies.

2

Fill the Table With Given Data

The given joint and marginal frequencies can now be added to the table.

two-way table

3

Find Any Missing Frequencies

Using the given frequencies, more information can potentially be found by reasoning. For instance, because out of the males prefer berets, the number of males who prefer top hats is equal to the difference between these two values. Therefore, there are males who prefer top hats. Since there are females who prefer top hats, the number of participants who prefer this type of hat is the sum of these two values. It has been found that participants prefer top hats. Continuing this reasoning, the entire table can be completed.

two-way table

Example

Night Owl or Early Bird?

As part of a school project, Zain asked people whether they are a night owl or an early bird. Zain decided to categorize the participants by their answer and age — younger than or aged or older. They obtained the following information.

  • people aged or older said they are early birds.
  • people younger than participated in the survey.
  • people said they are night owls.

Zain made a two-way frequency table and added this information.

two way table
Find the missing joint and marginal frequencies to help Zain complete the table!

Hint

Start by finding the number of people aged or older who participated in the survey. To do that, calculate the difference between the grand total and the number of participants younger than

Solution

Start by finding the missing marginal frequency of the last column of the table, labeled A. Note that people participated in the survey and of them are younger than Therefore, the number of participants aged or older can be found by calculating the difference between these two values. This information can be added to the table.

two way table

With this information, the joint frequency that represents the number of night owls aged or older can be calculated. Of the participants aged or older, are early birds. Therefore, the number of night owls aged or older is the difference between these two values. This information can also be added to the table.

two way table

The missing marginal frequency in the last row will now be calculated. Of the participants, said they are night owls. To find the number of early birds, the difference between these two values must be calculated. One more cell can be filled in!

two way table
Finally, the missing joint frequencies in the first row can be completed. The table can be completed with this information! Click on each cell to see its interpretation.
two way table

Discussion

Joint and Marginal Relative Frequencies

In a two-way frequency table, a joint relative frequency is the ratio of a joint frequency to the grand total. Similarly, a marginal relative frequency is the ratio of a marginal frequency to the grand total. Consider an example two-way table.

two-way table

Here, the grand total is The joint and marginal frequencies can now be divided by to obtain the and relative frequencies. Clicking in each cell will display its interpretation.

two-way table

Example

Joint and Marginal Relative Frequencies for Night Owls and Early Birds

Zain made a two-way frequency table as a part of a school project. The table categorizes the participants as night owls or early birds and as younger than years or aged or older.

two-way table representing the data collected by Zain

To get a deeper understanding of the preferences of the participants, Zain wants to calculate the joint and marginal relative frequencies.

two-way table to be completed with the joint and marginal relative frequencies
Help Zain complete the table!

Hint

Divide the joint and marginal frequencies by the grand total.

Solution

To obtain the joint and marginal relative frequencies, the joint and marginal frequencies must be divided by the grand total,

finding the joint and marginal relative frequencies

The table below shows the joint and marginal relative frequencies.

two-way table showing the joint and marginal relative frequencies

Discussion

Conditional Relative Frequency

A conditional relative frequency is the ratio of a joint frequency to either of its corresponding two marginal frequencies. Alternatively, it can be calculated using joint and marginal relative frequencies. As an example, the following data will be used.

two-way table

Using the column totals, the left column of joint frequencies should be divided by and the right column by Since the column totals are used, the sum of the conditional relative frequencies of each column is

two-way table

The resulting two-way frequency table can be interpreted to obtain the following information.

  • Out of all the participants with a driver's license, about of them own a car.
  • Out of all the participants with a driver's license, about of them do not own a car.
  • Out of all the participants without a driver's license, about of them own a car.
  • Out of all the participants without a driver's license, about of them do not own a car.

Example

Conditional Relative Frequencies for Night Owls and Early Birds

Using their two-way frequency table, Zain now wants to find the conditional relative frequencies.

two way table

To do so, Zain will use the row totals.

two way table
Help Zain to complete this table!

Hint

Since Zain uses the row totals, the joint frequencies in the first row must be divided by and the joint frequencies in the second row must be divided by

Solution

Zain uses the row totals. Therefore, the joint frequencies in the first row must be divided by and the joint frequencies in the second row must be divided by

two way table

The table below shows the conditional relative frequencies.

two way table

Example

Finding Conditional Probabilities Using Conditional Relative Frequencies

Zain will now consider the two-way table that shows conditional relative frequencies obtained using row totals.

two way table

They want to calculate some conditional probabilities by using the table. Help Zain find these probabilities!

a Knowing that a person is aged or older, find the probability that they are a night owl.
b Knowing that a person is younger than find the probability that they are an early bird.
c Knowing that a person is younger than find the probability that they are a night owl.
d Knowing that a person is aged or older, find the probability that they are an early bird.

Hint

Consider the fact that the conditional relative frequencies were found using row totals.

Solution

The table was created using row totals. Therefore, the first cell of the first row shows the probability of a person being a night owl given that they are younger than Similarly, the second cell of the first row shows the probability of a person being an early bird given that they are younger than

two way table

Likewise, the first cell of the second row shows the probability of a person being a night owl given that they aged or older. Similarly, the second cell of the second row shows the probability of a person being an early bird given that they are aged or older.

two way table

Example

Using a Two-Way Table to Determine Independence

Paulina conducted a survey at Washington High. She asked students whether they have cable TV and whether they took a vacation last summer. She displays the results in a two-way frequency table.

two-way table with categories took vacation and cable TV
Using the table, Paulina wants to find out whether or not taking a vacation and having cable TV are independent events for this population of students.

Hint

What is the probability that a student chosen at random took a vacation last summer? What is the probability that a random student who has cable TV took a vacation last summer?

Solution

Let be the event that a student took a vacation last summer and be the event that a student has cable TV. The table shows that from a total of participants, students took a vacation last summer.

two-way table
With this information, the probability of randomly choosing a student who took a vacation can be found.
Evaluate right-hand side
Next, the probability that a random student who has cable TV took a vacation last summer will be found. The table shows that out of the students who have cable TV, took a vacation last summer.
two way table
Now, the probability of event given event can be found.
Evaluate right-hand side
Comparing the found probabilities, it can be seen that they are not equal. This means that event a student having cable TV, affects event a student took a vacation last summer. Therefore, these events are not independent. Also, since students with cable TV are more likely to have taken a vacation last summer.

Closure

Chocolate Bar or Fruit?

At the beginning of the lesson, Zosia asked students of North High School in Honolulu whether they prefer a chocolate bar or a piece of fruit as a lunchtime snack and whether they surf or not.

surf, chocolate, and fruit
Letting be the event that a student surfs and the event that a student prefers a piece of fruit as a lunchtime snack, Zosia wants to calculate the following probabilities.

Hint

Make a two-way frequency table to display the obtained information.

Solution

A two-way frequency table can be made to organize the obtained information.

two way table

Next, the missing marginal frequencies can be calculated.

two way table

Now, two of the three missing joint frequencies can be calculated.

two way table

Finally, the last empty cell can be filled.

two way table

Now that the two-way table is complete, the desired probabilities can be found. Out of a total of students, surf and prefer fruit as a lunch snack.

two way table

With this information, and can be calculated. Also, of the students who prefer fruit, surf. Likewise, of the students who surf, prefer fruit.

two way table

Knowing this, and can be calculated.

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