. 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?
Well, it seems how much you make depends on how long you went to school.
To determine if the events "earn more than $100,000 per year" and "earned a bachelor's degree" are independent, we can compare the conditional probability to the marginal probability.
Let's denote the event "earn more than $100,000 per year" as A, and the event "earned a bachelor's degree" as B.
To determine if A and B are independent, we need to compare P(A) with P(A|B).
According to the problem, the probability that a randomly selected head of household in the United States earns more than $100,000 per year (P(A)) 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 (P(A|B)) is 0.412.
If A and B are independent events, then P(A|B) = P(A). In other words, the conditional probability of A given B should be equal to the marginal probability of A.
In this case, we can conclude that the events are NOT independent because P(A|B) = 0.412, which is not equal to P(A) = 0.202.
Therefore, the events "earn more than $100,000 per year" and "earned a bachelor's degree" are dependent.
To determine whether the events "earn more than $100,000 per year" and "earned a bachelor's degree" are independent, we need to compare the probability of both events occurring together with the probability of their individual occurrences.
Let's denote:
- A: The event that a randomly selected head of household earns more than $100,000 per year.
- B: The event that a randomly selected head of household has earned a bachelor's degree.
If the events A and B are independent, then the probability of both events occurring together (P(A ∩ B)) should equal the product of their individual probabilities (P(A) × P(B)).
Using the given information:
P(A) = 0.202 (the probability of earning more than $100,000 per year)
P(B) = ? (we need to calculate this)
P(A ∩ B) = 0.412 (the probability of earning more than $100,000 per year, given having a bachelor's degree)
Now we need to find P(B). We can use Bayes' theorem to do so:
P(A ∩ B) = P(B) × P(A|B)
Rearranging the formula, we have:
P(B) = P(A ∩ B) / P(A|B)
Plugging in the given values:
P(B) = 0.412 / P(A|B)
Now, we need to calculate P(A|B). This is the probability of earning more than $100,000 per year given that the head of household has a bachelor's degree.
To find P(A|B), we can use the formula:
P(A|B) = P(A ∩ B) / P(B)
Rearranging the formula, we have:
P(A|B) = P(A ∩ B) / P(B)
Plugging in the given values:
P(A|B) = 0.412 / P(B)
Now we have two equations:
1. P(B) = 0.412 / P(A|B)
2. P(A|B) = 0.412 / P(B)
To check if the events are independent, we compare P(A ∩ B) with P(A) × P(B).
If P(A ∩ B) = P(A) × P(B), the events are independent. Otherwise, they are dependent.
To summarize, to determine if the events "earn more than $100,000 per year" and "earned a bachelor's degree" are independent, we calculate P(B) using Bayes' theorem and compare P(A ∩ B) with P(A) × P(B). If the probabilities are equal, the events are independent; otherwise, they are dependent.