Answer :
Given the provided survey data, our task is to determine whether the events [tex]\( A \)[/tex] (the person has 2 siblings) and [tex]\( B \)[/tex] (the person does not have a pet) are independent events or not.
To do this, we need to calculate the following probabilities:
1. [tex]\( P(A) \)[/tex]: The probability that the person has 2 siblings.
2. [tex]\( P(B) \)[/tex]: The probability that the person does not have a pet.
3. [tex]\( P(A \mid B) \)[/tex]: The probability that the person has 2 siblings given that the person does not have a pet.
For [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent events, [tex]\( P(A \mid B) \)[/tex] should be equal to [tex]\( P(A) \)[/tex].
Step-by-Step Calculation:
1. Calculate [tex]\( P(A) \)[/tex]:
[tex]\[ P(A) = \frac{\text{Total number of people with 2 siblings}}{\text{Total number of survey participants}} = \frac{45}{250} = 0.18 \][/tex]
2. Calculate [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = \frac{\text{Total number of people who do not have a pet}}{\text{Total number of survey participants}} = \frac{100}{250} = 0.4 \][/tex]
3. Calculate [tex]\( P(A \mid B) \)[/tex]:
[tex]\[ P(A \mid B) = \frac{\text{Number of people with 2 siblings and no pet}}{\text{Total number of people who do not have a pet}} = \frac{18}{100} = 0.18 \][/tex]
4. Compare [tex]\( P(A \mid B) \)[/tex] and [tex]\( P(A) \)[/tex]:
[tex]\[ P(A \mid B) = 0.18 \quad \text{and} \quad P(A) = 0.18 \][/tex]
Since [tex]\( P(A \mid B) = P(A) = 0.18 \)[/tex], we can conclude that the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
Therefore, the correct statement is:
A and B are independent events because [tex]\( P(A \mid B) = P(A) = 0.18 \)[/tex].
To do this, we need to calculate the following probabilities:
1. [tex]\( P(A) \)[/tex]: The probability that the person has 2 siblings.
2. [tex]\( P(B) \)[/tex]: The probability that the person does not have a pet.
3. [tex]\( P(A \mid B) \)[/tex]: The probability that the person has 2 siblings given that the person does not have a pet.
For [tex]\( A \)[/tex] and [tex]\( B \)[/tex] to be independent events, [tex]\( P(A \mid B) \)[/tex] should be equal to [tex]\( P(A) \)[/tex].
Step-by-Step Calculation:
1. Calculate [tex]\( P(A) \)[/tex]:
[tex]\[ P(A) = \frac{\text{Total number of people with 2 siblings}}{\text{Total number of survey participants}} = \frac{45}{250} = 0.18 \][/tex]
2. Calculate [tex]\( P(B) \)[/tex]:
[tex]\[ P(B) = \frac{\text{Total number of people who do not have a pet}}{\text{Total number of survey participants}} = \frac{100}{250} = 0.4 \][/tex]
3. Calculate [tex]\( P(A \mid B) \)[/tex]:
[tex]\[ P(A \mid B) = \frac{\text{Number of people with 2 siblings and no pet}}{\text{Total number of people who do not have a pet}} = \frac{18}{100} = 0.18 \][/tex]
4. Compare [tex]\( P(A \mid B) \)[/tex] and [tex]\( P(A) \)[/tex]:
[tex]\[ P(A \mid B) = 0.18 \quad \text{and} \quad P(A) = 0.18 \][/tex]
Since [tex]\( P(A \mid B) = P(A) = 0.18 \)[/tex], we can conclude that the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent.
Therefore, the correct statement is:
A and B are independent events because [tex]\( P(A \mid B) = P(A) = 0.18 \)[/tex].
