Question

11 multiple choice 5 points
which of the following events are independent given p(a), p(b), and p(a and b)?
○ p(a) = 0.3; p(b) = 0.15; p(a and b) = 0.045
○ p(a) = 0.16; p(b) = 0.24; p(a and b) = 0.32
○ p(a) = 0.25; p(b) = 0.25; p(a and b) = 0.5
○ p(a) = 0.08; p(b) = 0.4; p(a and b) = 0.12

12 multiple choice 5 points
in a certain town, the probability that a person plays sports is 65%. the probability that a person is between the ages of 12 and 18 is 40%. the probability that a person plays sports and is between the ages of 12 and 18 is 26%.
○ these events are dependent
○ these events are independent

Explanation:

Question 11

Step1: Recall Independence Rule

Two events \( A \) and \( B \) are independent if \( P(A \cap B) = P(A) \times P(B) \).

Step2: Check Option 1

\( P(A) = 0.3 \), \( P(B) = 0.15 \), \( P(A \cap B) = 0.045 \).
Calculate \( P(A) \times P(B) = 0.3 \times 0.15 = 0.045 \).
This matches \( P(A \cap B) \), so check if others fail.

Step3: Check Option 2

\( P(A) = 0.16 \), \( P(B) = 0.24 \).
\( P(A) \times P(B) = 0.16 \times 0.24 = 0.0384
eq 0.32 \).

Step4: Check Option 3

\( P(A) = 0.25 \), \( P(B) = 0.25 \).
\( P(A) \times P(B) = 0.25 \times 0.25 = 0.0625
eq 0.5 \).

Step5: Check Option 4

\( P(A) = 0.08 \), \( P(B) = 0.4 \).
\( P(A) \times P(B) = 0.08 \times 0.4 = 0.032
eq 0.12 \).

Step1: Define Events

Let \( A \): plays sports (\( P(A) = 0.65 \)), \( B \): age 12–18 (\( P(B) = 0.40 \)).
\( P(A \cap B) = 0.26 \).

Step2: Check Independence

Calculate \( P(A) \times P(B) = 0.65 \times 0.40 = 0.26 \).
This equals \( P(A \cap B) \), so events are independent.

Answer:

The first option: \( P(A) = 0.3; P(B) = 0.15; P(A \text{ and } B) = 0.045 \)

Question 12