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. A and B are independent events because [tex]P(A | B) = P(A)=0.16[/tex].
B. A and B are independent events because [tex]P(A | B) = P(A)=0.75[/tex].
C. A and B are not independent events because [tex]P(A | B)=0.16[/tex] and [tex]P(A)=0.75[/tex].
D. A and B are not independent events because [tex]P(A | B)=0.75[/tex] and [tex]P(A)=0.16[/tex].
Answer (2)
Calculate P ( A ) : P ( A ) = 300 225 = 0.75 .
Calculate P ( A ∣ B ) : P ( A ∣ B ) = 48 36 = 0.75 .
Compare P ( A ) and P ( A ∣ B ) : 0.75 = 0.75 .
Conclude that A and B are independent events because P ( A ∣ B ) = P ( A ) = 0.75 .
Explanation
Analyze the problem Let's analyze the problem. We are given a table with data about people who have gone surfing (event A) and snowboarding (event B). We need to determine if events A and B are independent. Two events are independent if P ( A ∣ B ) = P ( A ) . We will calculate P ( A ) and P ( A ∣ B ) and compare them.
Calculate P(A) First, we calculate P ( A ) , the probability that a person has surfed. From the table, we know that 225 out of 300 people have surfed. Therefore, P ( A ) = 300 225 = 0.75
Calculate P(A|B) Next, we calculate P ( A ∣ B ) , the probability that a person has surfed given that they have snowboarded. From the table, we know that 36 people have surfed and snowboarded, and 48 people have snowboarded. Therefore, P ( A ∣ B ) = 48 36 = 0.75
Compare P(A) and P(A|B) Now, we compare P ( A ) and P ( A ∣ B ) . We found that P ( A ) = 0.75 and P ( A ∣ B ) = 0.75 . Since P ( A ∣ B ) = P ( A ) , events A and B are independent.
Conclusion Since P ( A ∣ B ) = P ( A ) = 0.75 , the correct statement is: A and B are independent events because P ( A ∣ B ) = P ( A ) = 0.75 .
Examples
In marketing, understanding the independence of events can help predict consumer behavior. For example, if liking a certain product (A) is independent of age group (B), marketing strategies can be applied uniformly across all age groups. If they are dependent, strategies must be tailored. Here, we found surfing and snowboarding are independent events, meaning knowing someone snowboards doesn't change the likelihood they surf. This might influence how a sports equipment company markets its products.
Events A (surfing) and B (snowboarding) are independent because the probabilities of A given B and A on its own are equal, both being 0.75. Therefore, the correct answer is Option A. This indicates that knowing someone has snowboarded doesn't change the likelihood of them having surfed.
;
