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.

a. Find $P\left(athletic\right|5thgrade)$.

b. Use your answer from part (a) to help determine if the events “$5th$ grade” and “athletic” are independent.

Short answer

Part a. Conditional probability,

$P\left(athletic\right|5thgrade)\approx 0.3333$

Part b. No, the events “$5th$ grade” and “athletic” are not independent.

Step-by-step solution

Part a. Step 1. Given information

Survey data summarized in two – way table:

Part a. Step 2. Calculation

According to conditional probability,

$P\left(B\right|A)=\frac{P(A\cap B)}{P\left(A\right)}=\frac{P(AandB)}{P\left(A\right)}$

Note that

The information about $335$students is provided in the table.

Thus,

The number of possible outcomes is $335$.

Also note that

In the table, $108$of the $335$students are $5^{th}$graders.

Thus,

The number of favorable outcomes is $108$.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

$P(5thgrade)=\frac{Numberoffavourableoutcomes}{Numberofpossibleoutcomes}=\frac{108}{335}$

Now,

Note that

In the table, $36$of the $335$students are $5^{th}$grades and think that athletics are most important. In this case, the number of favorable outcomes is $36$and number of possible outcomes is $335$.

$P(Athleticand5thgrade)=\frac{Numberoffavourableoutcomes}{Numberofpossibleoutcomes}=\frac{36}{335}$

Apply conditional probability:

$P\left(Athletic\right|5thgrade)=\frac{P(Athleticand5thgrade)}{P(5thgrade)}=\frac{{\displaystyle \frac{36}{335}}}{{\displaystyle \frac{108}{335}}}=\frac{36}{108}\approx 0.3333$

Thus,

The conditional probability for the randomly selected $5^{th}$grade student thinks athletic ability as most important is approx. $0.3333$.

Part b Step 1. Calculation

The two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

According to conditional probability,

$P\left(B\right|A)=\frac{P(A\cap B)}{P\left(A\right)}=\frac{P(AandB)}{P\left(A\right)}$

Note that

The information about $335$students is provided in the table.

Thus,

The number of possible outcomes is $335$.

Also note that

In the table, $108$of the $335$students are $5^{th}$graders.

Thus,

The number of favorable outcomes is $108$.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

$P(5thgrade)=\frac{Numberoffavourableoutcomes}{Numberofpossibleoutcomes}=\frac{108}{335}$

Also note that

In the table, $98$of the $335$students think athletics as most important.

Thus,

The number of favorable outcomes is $98$and the number of possible outcomes is $335$.

When the number of favorable outcomes is divided by the number of possible outcomes, we get the probability.

$P\left(Athletic\right)=\frac{Numberoffavourableoutcomes}{Numberofpossibleoutcomes}=\frac{98}{335}\approx 0.2925$

Now,

Note that

In the table, $36$of the $335$students are $5^{th}$grades and think that athletics are most important. In this case, the number of favorable outcomes is $36$and number of possible outcomes is $335$.

$P(Athleticand5thgrade)=\frac{Numberoffavourableoutcomes}{Numberofpossibleoutcomes}=\frac{36}{335}$

Apply conditional probability:

$P\left(Athletic\right|5thgrade)=\frac{P(Athleticand5thgrade)}{P(5thgrade)}=\frac{{\displaystyle \frac{36}{335}}}{{\displaystyle \frac{108}{335}}}=\frac{36}{108}\approx 0.3333$

For events “Athletic” and “$5^{th}$grade” to be independent,

$P\left(Athletic\right|5thgrade)=P(Athletic)$

And

$P(5thgrade|Atheltic)=P(5thgrade)$

But

We have

$P\left(Athletic\right|5thgrade)\approx 0.3333$

And

$P\left(athletic\right)\approx 0.2925$

Both probabilities are not identical.

Thus,

Both events are not independent.