A survey was taken of children between the ages of 3 and 7. Let A be the event that the person has 2 siblings, and let B be the event that the person does not have a pet. 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)=0.18[/tex].
B. A and B are independent events because [tex]P(A | B) = P(A)=0.4[/tex].
C. A and B are not independent events because [tex]P(A | B) =0.4[/tex] and [tex]P(A)=0.18[/tex].
D. A and B are not independent events because [tex]P(A | B) =0.18[/tex] and [tex]P(A)=0.4[/tex].
Answer (2)
Calculate P ( A ) as the number of children with 2 siblings divided by the total number of children: P ( A ) = 250 45 = 0.18 .
Calculate P ( A ∣ B ) as the number of children with 2 siblings and no pet divided by the number of children with no pet: P ( A ∣ B ) = 100 18 = 0.18 .
Compare P ( A ) and P ( A ∣ B ) .
Since P ( A ∣ B ) = P ( A ) = 0.18 , events A and B are independent: A and B are independent events because P ( A ∣ B ) = P ( A ) = 0.18 .
Explanation
Understand the problem and provided data We are given a table with data from a survey of children between the ages of 3 and 7. Event A is defined as the event that a person has 2 siblings, and event B is defined as the event that a person does not have a pet. We want to determine whether events A and B are independent.
Define the condition for independence Two events A and B are independent if P ( A ∣ B ) = P ( A ) . We need to calculate P ( A ∣ B ) and P ( A ) and compare them.
Calculate P(A) First, let's calculate P ( A ) , the probability that a person has 2 siblings. From the table, we see that there are 45 children with 2 siblings out of a total of 250 children surveyed. Therefore, P ( A ) = 250 45 .
Calculate P(B) Now, let's calculate P ( B ) , the probability that a person does not have a pet. From the table, we see that there are 100 children who do not have a pet out of a total of 250 children surveyed. Therefore, P ( B ) = 250 100 .
Calculate P(A|B) Next, we calculate P ( A ∣ B ) , the probability that a person has 2 siblings given that they do not have a pet. From the table, we see that there are 18 children with 2 siblings who do not have a pet. The total number of children who do not have a pet is 100. Therefore, P ( A ∣ B ) = 100 18 .
Compare P(A) and P(A|B) We have P ( A ) = 250 45 = 0.18 and P ( A ∣ B ) = 100 18 = 0.18 . Since P ( A ∣ B ) = P ( A ) = 0.18 , events A and B are independent.
State the conclusion Therefore, the correct statement is: A and B are independent events because P ( A ∣ B ) = P ( A ) = 0.18 .
Examples
Understanding independence between events is crucial in many real-world scenarios. For instance, in marketing, a company might want to know if a customer's purchase of one product is independent of their purchase of another. If the events are independent, it means that promoting one product won't necessarily affect the sales of the other. This helps in making informed decisions about marketing strategies and resource allocation.
Events A and B are independent because the probability of having 2 siblings given that a child does not have a pet is equal to the overall probability of having 2 siblings. Therefore, the correct answer is A: A and B are independent events because P ( A ∣ B ) = P ( A ) = 0.18 .
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