Question

question
a group of students at a high school took a standardized test. the number of students who passed or failed the exam is broken down by those who did or did not eat breakfast in the following table. determine whether not eating breakfast and failing the test are independent by filling out the blanks in the sentence below, rounding all probabilities to the nearest thousandth.

	did eat breakfast	didnt eat breakfast
failed	7	21

answer attempt 1 out of 2
since p(didnt eat breakfast | fail) =
and p(didnt eat breakfast) =
, the two results are
so the events are
.

Explanation:

Step1: Calculate total number of students who failed

To find the total number of students who failed, we add the number of students who did eat breakfast and failed (7) and those who didn't eat breakfast and failed (21). So, total failed = \(7 + 21 = 28\).

Step2: Calculate \(P(\text{didn't eat breakfast} | \text{fail})\)

The formula for conditional probability is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\). Here, \(A\) is "didn't eat breakfast" and \(B\) is "fail". The number of students who didn't eat breakfast and failed is 21, and total failed is 28. So, \(P(\text{didn't eat breakfast} | \text{fail}) = \frac{21}{28} = 0.75\).

Step3: Calculate total number of students who didn't eat breakfast

To find the total number of students who didn't eat breakfast, we add the number of students who didn't eat breakfast and passed (63) and those who didn't eat breakfast and failed (21). So, total didn't eat breakfast = \(63 + 21 = 84\).

Step4: Calculate total number of students

Total number of students is the sum of all students: passed (108 + 63 = 171) and failed (28), so total = \(171 + 28 = 199\)? Wait, no, wait. Wait, passed: 108 (did eat) + 63 (didn't eat) = 171. Failed: 7 (did eat) + 21 (didn't eat) = 28. Total students = 171 + 28 = 199? Wait, no, maybe I made a mistake. Wait, actually, to find \(P(\text{didn't eat breakfast})\), we need total number of students. Let's recalculate total students: did eat breakfast: 108 (passed) + 7 (failed) = 115. Didn't eat breakfast: 63 (passed) + 21 (failed) = 84. So total students = 115 + 84 = 199. Then \(P(\text{didn't eat breakfast}) = \frac{84}{199} \approx 0.422\)? Wait, no, wait, maybe the problem is to check independence, so we need to see if \(P(A|B) = P(A)\). Wait, but the first blank is \(P(\text{didn't eat breakfast} | \text{fail})\), which we calculated as 0.75. Then \(P(\text{didn't eat breakfast})\) is total didn't eat breakfast (84) divided by total students (108 + 7 + 63 + 21 = 199? Wait, 108 + 7 is 115, 63 + 21 is 84, 115 + 84 is 199. So \(P(\text{didn't eat breakfast}) = \frac{84}{199} \approx 0.422\)? Wait, but that contradicts. Wait, maybe I misread the table. Let's check the table again:

| | Did Eat Breakfast | Didn't Eat Breakfast |
|----------|-------------------|----------------------|
| Passed | 108 | 63 |
| Failed | 7 | 21 |

So total did eat breakfast: 108 + 7 = 115. Total didn't eat breakfast: 63 + 21 = 84. Total students: 115 + 84 = 199.

Now, \(P(\text{didn't eat breakfast}) = \frac{84}{199} \approx 0.422\). But \(P(\text{didn't eat breakfast} | \text{fail}) = \frac{21}{28} = 0.75\). Wait, but the problem says "determine whether not eating breakfast and failing the test are independent". For independence, \(P(A|B) = P(A)\). So if \(P(\text{didn't eat breakfast} | \text{fail}) = P(\text{didn't eat breakfast})\), then they are independent. But here, 0.75 vs ~0.422, so they are not independent. But the question is to fill in the blanks: \(P(\text{didn't eat breakfast} | \text{fail})\) and \(P(\text{didn't eat breakfast})\).

Wait, maybe I made a mistake in total students. Wait, maybe the total number of students is 108 + 7 + 63 + 21 = 199? Wait, 108 + 7 is 115, 63 + 21 is 84, 115 + 84 is 199. So \(P(\text{didn't eat breakfast}) = \frac{84}{199} \approx 0.422\). But \(P(\text{didn't eat breakfast} | \text{fail}) = \frac{21}{28} = 0.75\). So the two results are 0.75 and approximately 0.422, so they are not equal, hence events are not independent. But the problem says "rounding all probabilities to the nearest thousandth". Wait, maybe I miscalculated total students. Wait, maybe the table is:

Passed: Did Eat Breakfast: 108, Didn't Eat Breakfast: 63. So passed total: 108 + 63 = 171.

Failed: Did Eat Breakfast: 7, Didn't Eat Breakfast: 21. So failed total: 7 + 21 = 28.

Total students: 171 + 28 = 199.

Now, \(P(\text{didn't eat breakfast}) = \frac{63 + 21}{199} = \frac{84}{199} \approx 0.422\) (rounded to nearest thousandth: 0.422).

\(P(\text{didn't eat breakfast} | \text{fail}) = \frac{21}{28} = 0.750\) (since 21/28 = 0.75, which is 0.750 when rounded to nearest thousandth).

So the first blank is 0.750, the second blank is approximately 0.422. Then, since 0.750 ≠ 0.422, the events are not independent.

Wait, but let's check again. Maybe the total number of students is calculated as (108 + 7) + (63 + 21) = 115 + 84 = 199. Correct.

So:

\(P(\text{didn't eat breakfast} | \text{fail}) = \frac{\text{number of students who didn't eat breakfast and failed}}{\text{total number of students who failed}} = \frac{21}{28} = 0.750\) (since 21 divided by 28 is 0.75, which is 0.750 when rounded to the nearest thousandth).

\(P(\text{didn't eat breakfast}) = \frac{\text{total number of students who didn't eat breakfast}}{\text{total number of students}} = \frac{63 + 21}{108 + 7 + 63 + 21} = \frac{84}{199} \approx 0.422\) (rounded to the nearest thousandth: 0.422).

Since 0.750 ≠ 0.422, the two events are not independent.

Answer:

First blank: \(0.750\), Second blank: \(0.422\), and the events are not independent.