Question

The two-way table summarizes data on whether students at a certain high school eat

regularly in the school cafeteria by grade level.

a. If you choose a student at random, what is the probability that the student eats

regularly in the cafeteria and is not a $10th‐$grader?

b. If you choose a student at random who eats regularly in the cafeteria, what is the probability that the student is a $10th‐$grader?

c. Are the events “$10th‐$grader” and “eats regularly in the cafeteria” independent?

Justify your answer.

Short answer

a. the probability that the student eats regularly in the cafeteria and is not a$10th‐$ grader is$39.75\%$

b. the probability that a student at random who eats regularly in the cafeteria and is a$10th$$‐$grader

is $35.35\%$

c. the events “10th-grader” and “eats regularly in the cafeteria” are not independent.

Step-by-step solution

Part (a): Step 1: Given information

We have been given a two-way table that summarizes data on whether students at a certain high school eat regularly in the school cafeteria by grade level.

We need to find out the probability that the student eats regularly in the cafeteria and is not a $10th‐$grader.

Part (a): Step 2: Explanation

Let A$=$Student who eats regularly in the cafeteria and is a $10th‐$grader.

Let B$=$Student who eats regularly in the cafeteria and is not a $10th‐$grader.

$$\begin{gathered} P\left(B\right)=\frac{no.offavourableoutcomes}{no.ofpossibleoutcomes} \\ =\frac{320}{805}=\frac{64}{161}\approx 0.3975=39.75\% \end{gathered}$$

Part (b): Step 1: Given information

We have been given a two-way table that summarizes data on whether students at a certain high school eat regularly in the school cafeteria by grade level.

We need to find out the probability that the student eats regularly in the cafeteria and is a $10th‐$grader.

Part (b): Step 2: Explanation

Let A$=$Student who eats regularly in the cafeteria and is a $10th‐$grader.

Let B $=$Student who eats regularly in the cafeteria and is not a $10th‐$grader.

$$\begin{gathered} P\left(A\right)=\frac{no.offavourableoutcomes}{no.ofpossibleoutcomes} \\ =\frac{175}{495}=\frac{35}{99}\approx .3535=35.35\% \end{gathered}$$

Part (c): Step 1: Given information

We have been given a two-way table that summarizes data on whether students at a certain high school eat regularly in the school cafeteria by grade level.

We need to find out whether the events “$10th‐$grader” and “eats regularly in the cafeteria” are independent or not.

Part (c): Step 2: Explanation

Let A$=$Student who eats regularly in the cafeteria and is a $10th‐$grader.

$$\begin{gathered} P\left(A\right)=\frac{no.offavourableoutcomes}{no.ofpossibleoutcomes} \\ =\frac{175}{495}=\frac{35}{99}\approx .3535=35.35\% \end{gathered}$$

Let C$=$Students of $10th‐$grade

$$\begin{gathered} P\left(C\right)=\frac{no.offavourableoutcomes}{no.ofpossibleoutcomes} \\ =\frac{209}{805}\approx 0.2596=25.96\% \end{gathered}$$

As shown above,$P\left(A\right)\ne P\left(C\right)$which implies that the events “$10th‐$grader” and “eats regularly in the cafeteria” are not independent