# Two-Way Tables and Probability

Alignments to Content Standards: S-CP.A.4

Each student in a random sample of seniors at a local high school participated in a survey. These students were asked to indicate their gender and their eye color. The following table summarizes the results of the survey.

 Eye color Brown Blue Green Total Gender Male 50 40 20 110 Female 40 40 10 90 Total 90 80 30 200

1. Suppose that one of these seniors is randomly selected. What is the probability that the selected student is a male?

2. Suppose that one of these seniors is randomly selected. What is the probability that the selected student has blue eyes?

3. Suppose that one of these seniors is randomly selected. What is the probability that the selected student is a male and has blue eyes?

4. Suppose that one of these seniors is randomly selected. What is the probability that the selected student is a male or has blue eyes?

5. Suppose that one of these seniors is randomly selected. What is the probability that the selected student has blue eyes, given that the student is male?

6. Suppose that one of these seniors is randomly selected. What is the probability that the selected student is a male, given that the student has blue eyes?

## IM Commentary

The purpose of this task is to provide practice using data in a two-way table to calculate probabilities, including conditional probabilities (S.CP.4). This task provides a good starting point for a discussion of conditional probabilities leading to the concept of independence. In particular, focus on the difference between the two conditional probabilities calculated in parts (e) and (f).

## Solution

a. ${number \ of \ males \over number \ of \ students}={110 \over 200}=0.55$

b.  ${number \ of \ students \ with \ blue \ eyes \over number \ of \ students}={80 \over 200}=0.40$

c.  ${number \ of \ males \ with \ blue \ eyes\over number \ of \ students}={40 \over 200}=0.20$

d.  ${number \ of \ males \ + \ number \ of \ blue \ eyed \ females\over number \ of \ students}={110 \ + \ 40 \over 200}=0.75$

OR

${number \ of \ males \ + \ number \ of \ blue \ eyed \ students \ - \ number \ of \ blue \ eyed \ boys \over number \ of \ students}={110 \ + \ 80 \ - \ 40 \over 200}=0.75$

e.  ${number \ of \ blue \ eyed \ males \over number \ of \ males}={40 \over 110}=0.36$

f.  ${number \ of \ blue \ eyed \ males \over number \ of \ blue \ eyed \ students}={40 \over 80}=0.50$