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 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	2	68

answer attempt 1 out of 2
since p(did eat breakfast | fail) = \boxed{} and p(did eat breakfast) = \boxed{}, the two results are \boxed{\text{so the events are}}
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Explanation:

Step1: Calculate total students who failed

To find the number of students who failed, we sum the number of students who failed and ate breakfast and those who failed and didn't eat breakfast. So, \( 2 + 68 = 70 \).

Step2: Calculate \( P(\text{did eat breakfast} | \text{fail}) \)

The formula for conditional probability is \( P(A|B)=\frac{P(A\cap B)}{P(B)} \). Here, \( A \) is "did eat breakfast" and \( B \) is "failed". The number of students who failed and ate breakfast is 2, and the total number of students who failed is 70. So, \( P(\text{did eat breakfast} | \text{fail})=\frac{2}{70}\approx0.029 \) (rounded to the nearest thousandth).

Step3: Calculate total number of students

We sum all the values in the table: \( 74 + 19 + 2 + 68 = 163 \).

Step4: Calculate \( P(\text{did eat breakfast}) \)

The number of students who ate breakfast is \( 74 + 2 = 76 \). So, \( P(\text{did eat breakfast})=\frac{76}{163}\approx0.466 \) (rounded to the nearest thousandth).

Step5: Compare the two probabilities

Since \( 0.029
eq0.466 \), the events "did eat breakfast" and "failed" are not independent (because for independent events, \( P(A|B) = P(A) \)).

Answer:

\( P(\text{did eat breakfast} | \text{fail}) \approx 0.029 \), \( P(\text{did eat breakfast}) \approx 0.466 \), and the events are not independent.