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 passing 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	21	30

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

Explanation:

Step1: Calculate total students who ate breakfast

Total students who ate breakfast: \( 70 + 21 = 91 \)

Step2: Calculate total students who passed

Total students who passed: \( 70 + 100 = 170 \)

Step3: Calculate total number of students

Total students: \( 91 + 100 + 30 = 221 \) (Wait, no: \( 70 + 21 + 100 + 30 = 221 \))

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

\( P(\text{did eat breakfast}) = \frac{91}{221} \approx 0.41176 \)

Step5: Calculate \( P(\text{pass}) \)

\( P(\text{pass}) = \frac{170}{221} \approx 0.76923 \)

Step6: Calculate \( P(\text{did eat breakfast}) \times P(\text{pass}) \)

\( 0.41176 \times 0.76923 \approx 0.317 \) (rounded to nearest thousandth)

Step7: Calculate \( P(\text{did eat breakfast and pass}) \)

\( P(\text{did eat breakfast and pass}) = \frac{70}{221} \approx 0.31674 \approx 0.317 \) (rounded to nearest thousandth)

Answer:

First blank: \( 0.317 \), Second blank: \( 0.317 \) (so the events are independent as the two probabilities are equal when rounded to nearest thousandth)