Chapter 5.3, Problem 74E

(a)

To determine

Conditional probability for the 5^th grade student thinksathletic ability as most important.

Answer to Problem 74E

Conditional probability,

  $P\left(\text{Athletic}|\text{5th}\text{grade}\right)\approx 0.3333$

Explanation of Solution

Given information:

Survey data summarized in two − way table:

  

Calculations:

According to conditional probability,

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

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.

  $P\left(\text{5th}\text{grade}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{108}{335}$

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.

  $P\left(\text{Athletic}\text{and}\text{5th}\text{grade}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{36}{335}$

Apply conditional probability:

  $P\left(\text{Athletic}|\text{5th}\text{grade}\right)=\frac{P\left(\text{Athetic}\text{and}\text{5th}\text{grade}\right)}{P\left(\text{5th}\text{grade}\right)}=\frac{\frac{36}{335}}{\frac{108}{335}}=\frac{36}{108}\approx 0.3333$

Thus,

The conditional probability for the randomly selected 5^th grade student thinksathletic ability as most important is approx. 0.3333.

(b)

To determine

Whether the events “5^th grade” and “athletic” independent.

Answer to Problem 74E

No, the events “5^th grade” and “athletic” are not independent.

Explanation of Solution

Given information:

Survey data summarized in two − way table:

  

Calculations:

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,

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

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.

  $P\left(\text{5th}\text{grade}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{108}{335}$

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.

  $P\left(\text{Athletic}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{98}{335}\approx 0.2925$

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.

  $P\left(\text{Athletic}\text{and}\text{5th}\text{grade}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{outcomes}}{\text{Number}\text{of}\text{possible}\text{outcomes}}=\frac{36}{335}$

Apply conditional probability:

  $P\left(\text{Athletic}|\text{5th}\text{grade}\right)=\frac{P\left(\text{Athetic}\text{and}\text{5th}\text{grade}\right)}{P\left(\text{5th}\text{grade}\right)}=\frac{\frac{36}{335}}{\frac{108}{335}}=\frac{36}{108}\approx 0.3333$

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

  $P\left(\text{Athletic}|\text{5th}\text{grade}\right)=P\left(\text{Athletic}\right)$

And

  $P\left(\text{5th}\text{grade}|\text{Athletic}\right)=P\left(\text{5th}\text{grade}\right)$

But

We have

  $P\left(\text{Athletic}|\text{5th}\text{grade}\right)\approx 0.3333$

And

  $P\left(\text{Athletic}\right)\approx 0.2925$

Both probabilities are not identical.

Thus,

Both events are not independent.