*65*

A **two-way frequency table** is a table that displays the frequencies (or “counts”) for two categorical variables.

For example, the following two-way table shows the results of a survey that asked 100 people which sport they liked best: baseball, basketball, or football. The rows display the gender of the respondent and the columns show which sport they chose:

This is a *two-way *table because we have two categorical variables: *gender* and *favorite sport*.

The numbers in the body of the table are called **joint frequencies** and the numbers that display the total row and column frequencies are called **marginal frequencies**.

Here is how to interpret this table:

- A total of 100 people responded to this survey.
- Of the 100 total respondents, 48 were males and 52 were females.
- A total of 36 respondents said they like baseball the most, 31 said they like basketball the most, and 33 said they like football the most.
- A total of 13 males said they like baseball the most, 23 females said they like baseball the most, 15 males said they like basketball the most, 16 females said they like basketball the most, 20 males said they like football the most, and 13 females said they like football the most.

**How to Find Conditional Relative Frequencies Using a Two-Way Table**

A two-way frequency table is useful for helping us find **conditional relative frequencies**. These are frequencies that are based on some *condition*.

The following examples illustrate how to use a two-way frequency table to find conditional relative frequencies.

**Example 1**

What is the probability that a survey respondent likes basketball the most, *given that the respondent is male*?

Since we are given the condition that the respondent is male, we only want to look at the row that contains the male responses. To find the probability that the respondent likes basketball the most, we can simply divide the number of male respondents who like basketball the most by the total number of males:

Thus, the probability that a survey respondent likes basketball the most, *given that the respondent is male* is 0.3125, or **31.25%**.

**Example 2**

What is the probability that a survey respondent likes baseball the most, *given that the respondent is female*?

Since we are given the condition that the respondent is female, we only want to look at the row that contains the female responses. To find the probability that the respondent likes baseball the most, we can simply divide the number of female respondents who like baseball the most by the total number of females:

Thus, the probability that a survey respondent likes baseball the most, *given that the respondent is female* is 0.4423, or **44.23%**.

**Example 3**

What is the probability that a survey respondent is male, *given that the respondent likes football the most*?

Since we are given the condition that the respondent likes football the most, we only want to look at the column that contains the responses of people who like football the most. To find the probability that the respondent is male, we can simply divide the number of males who like football the most by the total number of respondents who like football the most:

Thus, the probability that a survey respondent is male, *given that the respondent likes football the most** *is 0.606, or **60.6%**.

**Example 4**

What is the probability that a survey respondent is female, *given that the respondent likes baseball the most*?

Since we are given the condition that the respondent likes baseball the most, we only want to look at the column that contains the responses of people who like baseball the most. To find the probability that the respondent is female, we can simply divide the number of females who like baseball the most by the total number of respondents who like baseball the most:

Thus, the probability that a survey respondent is female, *given that the respondent likes baseball the most** *is 0.6389, or **63.89%**.

**Example 5**

What is the probability that a survey respondent likes baseball *or *football the most, *given that the respondent is male*?

Since we are given the condition that the respondent is male, we only want to look at the row that contains the responses of males. To find the probability that the respondent likes baseball *or *football the most, we can simply divide the number of males who like baseball or football the most by the total number of male respondents:

Thus, the probability that a survey respondent likes baseball *or *football the most, *given that the respondent is male** *is 0.6875, or **68.75%**.

**Example 6**

What is the probability that a survey respondent likes baseball *or *basketball the most, *given that the respondent is female*?

Since we are given the condition that the respondent is female, we only want to look at the row that contains the responses of females. To find the probability that the respondent likes baseball *or *basketball the most, we can simply divide the number of females who like baseball or basketball the most by the total number of female respondents:

Thus, the probability that a survey respondent likes baseball *or *basketball the most, *given that the respondent is female** *is 0.75, or **75%**.

**Example 7**

What is the probability that a survey respondent does *not *like football the most, *given that the respondent is male*?

Since we are given the condition that the respondent is male, we only want to look at the row that contains the responses of males. To find the probability that the respondent does not like football the most, we can simply divide the number of males who like baseball or basketball the most by the total number of male respondents:

Thus, the probability that a survey respondent does *not *like football the most, *given that the respondent is male** *is 0.5833, or **58.33%**.