Before you read

Answer first — see what you already know.

A high school surveyed 200 students about their preferred after-school activity and grade level:\n\n| | Sports | Arts | Total |\n|---|---|---|---|\n| Juniors | 50 | 60 | 110 |\n| Seniors | 40 | 50 | 90 |\n| Total | 90 | 110 | 200 |\n\nWhat percentage of all students prefer sports?

A
25%
B
45%
C
50%
D
55.6%

Check your answer

Answer:

Before you read

Answer first — see what you already know.

Using the same table: Of the seniors, what fraction prefer arts?

A
$\frac{50}{200}$
B
$\frac{50}{110}$
C
$\frac{50}{90}$
D
$\frac{90}{200}$

Check your answer

Answer:

Theory

Reading Two-Way Tables

A two-way table (contingency table) displays data for two categorical variables.

	Coffee	Tea	Total
Male	30	20	50
Female	25	25	50
Total	55	45	100

Joint frequency: a specific cell (e.g., Males who drink coffee = 30)
Marginal frequency: row or column total (e.g., Total males = 50)
Grand total: bottom-right cell (100)

Two-way table showing survey responses by gender

Example 1

Using the table above, what fraction of all people are females who drink tea?

Females who drink tea = 25

Total people = 100

Fraction $= \frac{25}{100} = 0.25$ or 25%

25%

Theory

Relative Frequency and Conditional Probability

Relative frequency $= \frac{\text{frequency}}{\text{total}}$

Three types:
1. Joint relative frequency: cell $\div$ grand total ("what fraction of everyone...")
2. Marginal relative frequency: row/column total $\div$ grand total
3. Conditional relative frequency: cell $\div$ row or column total ("given that..., what fraction...")

Conditional frequency answers questions like: "Of the males, what fraction drink coffee?" $\to$ $\frac{30}{50} = 60\%$

Example 1

Given that a person drinks coffee, what is the probability they are female?

Total coffee drinkers = 55

Female coffee drinkers = 25

Probability $= \frac{25}{55} = \frac{5}{11} \approx 0.45$

\frac{5}{11} \approx 45\%

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2 more sections including examples, practice problems, and step-by-step solutions.

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