Question

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.

Suppose we select one of these students at random. What’s the probability of each of the following?

a. The student is a sixth-grader or rated good grades as important.

b. The student is not a sixth-grader and did not rate good grades as important.

Short answer

Part a. Probability,

$P(6^{th}gradeorGrades)\approx 0.6985$

Part b. Probability,

$P(Not{6}^{th}graderandnoGrades)\approx 0.3015$

Step-by-step solution

Part a. Step 1. Given information

Survey data:

Part a. Step 2. Explanation

Look at the bottom right corner of the table,

There are $335$students in total.

Thus,

The number of possible outcomes is $335$.

Also,

For the student who are rated good grades or 6^th grader,

Add up all the values in column “Grades” and row “6th grade”, we will get $234$in total.

Thus,

The number of favorable outcomes is $234$.

Now,

The number of favorable outcomes divided by the number of possible outcomes gives the probability.

$$\begin{gathered} P(6^{th}gradeorgrades)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes} \\ =\frac{234}{335} \\ =0.6985 \end{gathered}$$

Part b. Step 1. Explanation

Look at the bottom right corner of the table,

There are $335$students in total.

Thus,

The number of possible outcomes is $335$.

Also,

For the student who did not rate good grades and not a 6^th grader as well,

Add up all the values of 4^th and 5^th grades in the columns “Athletic” and “Popular”, we will get $101$in total.

Thus,

The number of favorable outcomes is $101$.

Now,

The number of favorable outcomes divided by the number of possible outcomes gives the probability.

$$\begin{gathered} P(Not{6}^{th}graderandnogrades)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes} \\ =\frac{101}{335} \\ \approx 0.3015 \end{gathered}$$