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	9	42

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

Explanation:

Step1: Calculate total failed students

Total failed students = 9 (did eat) + 42 (didn't eat) = 51.

Step2: Find P(did eat breakfast | fail)

Using conditional probability formula \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), here \( A \) is "did eat breakfast", \( B \) is "failed". \( P(A \cap B) = 9 \), \( P(B) = 51 \). So \( P(\text{did eat} | \text{fail}) = \frac{9}{51} \approx 0.176 \).

Step3: Calculate total students who did eat breakfast

Total who did eat = 37 (passed) + 9 (failed) = 46.
Total students = 37 + 9 + 45 + 42 = 133.

Step4: Find P(did eat breakfast)

\( P(\text{did eat}) = \frac{46}{133} \approx 0.346 \).

Step5: Compare probabilities

Since \( 0.176
eq 0.346 \), the events are not independent.

Answer:

\( P(\text{did eat breakfast} | \text{fail}) \approx 0.176 \), \( P(\text{did eat breakfast}) \approx 0.346 \), and the events are not independent.