. According to the U.S. Census Bureau, the probability that a randomly selected head of household in the United States earns more than $100,000 per year is 0.202. The probability that a randomly selected head of household in the United States earns more than $100,000 per year, given that the head of household has earned a bachelor’s degree, is 0.412. Are the events “earn more than $100,000 per year” and “earned a bachelor’s degree” independent?
No, bachelor's degree group is part of total.
To determine if the events "earn more than $100,000 per year" and "earned a bachelor's degree" are independent, we need to compare the conditional probability with the marginal probability.
Let's break it down:
Step 1: Calculate the probability of earning more than $100,000 per year (P(A)). According to the U.S. Census Bureau, this probability is 0.202.
Step 2: Calculate the probability of having earned a bachelor's degree (P(B)). We don't have this information, so let's assume it is X.
Step 3: Calculate the joint probability of both events occurring: earning more than $100,000 per year and having earned a bachelor's degree (P(A ∩ B)). We don't have this information, so let's assume it is Y.
Step 4: Calculate the conditional probability of earning more than $100,000 per year given that the head of household has earned a bachelor's degree (P(A|B)). According to the information given, this probability is 0.412.
To determine if the events are independent, we compare P(A|B) with P(A). If the two probabilities are approximately equal, then the events are considered independent. Mathematically, if P(A|B) ≈ P(A), then the events are independent.
In this case, P(A) = 0.202 and P(A|B) = 0.412.
Since P(A|B) is not approximately equal to P(A), we conclude that the events "earn more than $100,000 per year" and "earned a bachelor's degree" are dependent.