Two-Way Tables and Probability

Task

 

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.  \({number \ of \ blue \ eyed \ males \over number \ of \ blue \ eyed \ students}={40 \over 80}=0.50\)