Alejandro surveyed his classmates to determine who has ever gone surfing and who has ever gone snowboarding. Let [tex]$A$[/tex] be the event that the person has gone surfing, and let [tex]$B$[/tex] be the event that the person has gone snowboarding.
| | Has Snowboarded | Never Snowboarded | Total |
| :------------- | :--------------- | :----------------- | :---- |
| Has Surfed | 36 | 189 | 225 |
| Never Surfed | 12 | 63 | 75 |
| Total | 48 | 252 | 300 |
Which statement is true about whether [tex]$A$[/tex] and [tex]$B$[/tex] are independent events?
A. [tex]$A$[/tex] and [tex]$B$[/tex] are independent events because [tex]$P(A | B) = P(A) = 0.16$[/tex]
B. [tex]$A$[/tex] and [tex]$B$[/tex] are independent events because [tex]$P(A | B) = P(A) = 0.75$[/tex]
C. [tex]$A$[/tex] and [tex]$B$[/tex] are not independent events because [tex]$P(A | B) = 0.16$[/tex] and [tex]$P(A) = 0.75$[/tex]
D. [tex]$A$[/tex] and [tex]$B$[/tex] are not independent events because [tex]$P(A \cdot A) = 0.75$[/tex] and [tex]$P(A) = 0.16$[/tex]
Answer (1)
Calculate P ( A ) as the number of students who have surfed divided by the total number of students: P ( A ) = 300 225 = 0.75 .
Calculate P ( A ∣ B ) as the number of students who have surfed and snowboarded divided by the number of students who have snowboarded: P ( A ∣ B ) = 48 36 = 0.75 .
Compare P ( A ) and P ( A ∣ B ) . Since P ( A ∣ B ) = P ( A ) = 0.75 , the events A and B are independent.
Conclude that A and B are independent events because P ( A / B ) = P ( A ) = 0.75 . 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 that shows the results of a survey about surfing and snowboarding habits of Alejandro's classmates. We need to determine if the events 'has gone surfing' (A) and 'has gone snowboarding' (B) are independent. Two events are independent if the probability of one event occurring is not affected by whether the other event has occurred. Mathematically, A and B are independent if P ( A ∣ B ) = P ( A ) .
Calculate P(A) First, we need to calculate P ( A ) , the probability that a randomly selected classmate has gone surfing. From the table, we see that 225 out of 300 students have gone surfing. Therefore, P ( A ) = 300 225 = 0.75
Calculate P(A|B) Next, we need to calculate P ( A ∣ B ) , the probability that a randomly selected classmate has gone surfing given that they have gone snowboarding. From the table, we see that 36 students have gone surfing and snowboarding, and 48 students have gone snowboarding. 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 ) , the events A and B are independent.
Conclusion Therefore, the correct statement is: A and B are independent events because P ( A / B ) = P ( A ) = 0.75 .
Examples
In market research, determining the independence of events like 'customer buys product A' and 'customer buys product B' can help businesses understand if marketing strategies for one product affect the sales of another. If the events are independent, it suggests that the products are not related in the customer's mind, and separate marketing strategies would be more effective. If they are dependent, a joint marketing campaign might be more beneficial.
