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	18	36

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

Explanation:

Step1: Calculate total students who ate breakfast

Total students who ate breakfast: \(50 + 18 = 68\)

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

Number of students who ate breakfast and failed: \(18\)
\(P(\text{fail} | \text{did eat breakfast})=\frac{18}{68}\approx0.265\)

Step3: Calculate total students

Total students: \((50 + 18)+(100 + 36)=204\)

Step4: Find \(P(\text{fail})\)

Number of students who failed: \(18 + 36 = 54\)
\(P(\text{fail})=\frac{54}{204}\approx0.265\)

Step5: Compare probabilities

Since \(P(\text{fail} | \text{did eat breakfast})\approx0.265\) and \(P(\text{fail})\approx0.265\), they are equal.

Answer:

\(P(\text{fail} | \text{did eat breakfast}) \approx 0.265\), \(P(\text{fail}) \approx 0.265\), the two results are equal, so the events are independent.