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:

Step1: Calculate \(P(A)\)

The formula for probability is \(P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
The total number of people surveyed is \(n = 300\), and the number of people who have surfed (\(A\)) is \(n(A)=225\).
So, \(P(A)=\frac{225}{300}=0.75\).

Step2: Calculate \(P(A|B)\)

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).
\(P(A\cap B)\) is the probability that a person has surfed and snow - boarded. \(n(A\cap B) = 36\), so \(P(A\cap B)=\frac{36}{300}\).
\(P(B)\) is the probability that a person has snow - boarded. \(n(B)=48\), so \(P(B)=\frac{48}{300}\).
Then \(P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{\frac{36}{300}}{\frac{48}{300}}=\frac{36}{48}=0.75\).

Answer:

A and B are independent events because \(P(A|B)=P(A) = 0.75\).