Question

alejandro surveyed his classmates to determine who has ever gone surfing and who has ever gone snowboarding. let a be the event that the person has gone surfing, and let b be the event that the person has gone snowboarding.

which statement is true about whether a and b are independent events?

a and b are independent events because \\(p(a|b) = p(a) = 0.16\\).

a and b are independent events because \\(p(a|b) = p(a) = 0.75\\).

a and b are not independent events because \\(p(a|b) = 0.16\\) and \\(p(a) = 0.75\\).

a and b are not independent events because \\(p(a|b) = 0.75\\) and \\(p(a) = 0.16\\).

Explanation:

Calculate the probability of event A

Using the Two-Way Frequency Tables knowledge point

$$ P(A) = \frac{\text{Total Surfed}}{\text{Grand Total}} = \frac{225}{300} = 0.75 $$

Calculate the conditional probability of A given B

Using the Conditional Probability Calculation knowledge point

$$ P(A|B) = \frac{\text{Has Surfed and Has Snowboarded}}{\text{Total Has Snowboarded}} = \frac{36}{48} = 0.75 $$

Determine independence of events A and B

Using the Independent Events knowledge point

$$
LATEXBLOCK0
$$

Answer:

• (A) A and B are independent events because \(P(A|B) = P(A) = 0.16\).
• (B) A and B are independent events because \(P(A|B) = P(A) = 0.75\). (Correct answer)
• (C) A and B are not independent events because \(P(A|B) = 0.16\) and \(P(A) = 0.75\).
• (D) A and B are not independent events because \(P(A|B) = 0.75\) and \(P(A) = 0.16\).