Marriage and Education
According to the U.S. Census Bureau, 20.2% of American women aged 25 years or older have a Bachelor's Degree. 16.5% have never married. Among women 25 years or older who have never married, 22.8% have a Bachelor's Degree. Among women 25 years or older who have a Bachelor's Degree, 18.6% have never married.
a. Are the events "have a Bachelor's Degree" and "never married" independent? Explain.

Verified step by step guidance

1. Identify the two events: Let A = "have a Bachelor's Degree" and B = "never married." We are given the probabilities: P(A) = 0.202, P(B) = 0.165, P(A | B) = 0.228, and P(B | A) = 0.186.
2. Recall the definition of independence: Two events A and B are independent if and only if P(A \cap B) = P(A) \times P(B). Alternatively, independence means P(A | B) = P(A) and P(B | A) = P(B).
3. Calculate P(A \cap B) using the conditional probability P(A | B): Use the formula P(A \cap B) = P(A | B) \times P(B). Substitute the values to express P(A \cap B) in terms of the given probabilities.
4. Compare P(A \cap B) with P(A) \times P(B): Calculate P(A) \times P(B) and check if it equals the value of P(A \cap B) found in the previous step.
5. Based on the comparison, conclude whether the events are independent: If P(A \cap B) = P(A) \times P(B), then A and B are independent; otherwise, they are dependent. Also, verify if P(A | B) equals P(A) or if P(B | A) equals P(B) to support your conclusion.

Key Concepts

Probability of Events

Probability measures the likelihood of an event occurring, expressed as a value between 0 and 1. Understanding how to calculate and interpret probabilities for single and combined events is essential for analyzing relationships between different characteristics, such as education level and marital status.

Independence of Events

Two events are independent if the occurrence of one does not affect the probability of the other. Mathematically, events A and B are independent if P(A ∩ B) = P(A) × P(B). Checking this condition helps determine whether having a Bachelor's Degree and never marrying influence each other.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has occurred, denoted P(A|B). It helps compare probabilities within subgroups, such as the proportion of women with a Bachelor's Degree among those who never married, to assess dependence or independence between events.