calculate conditional probability of swim team members being philosophy majors
Check the final answer first, then review the worked steps.
Answer
0.5
Step-by-step solution
- Identify the given probabilities: Let $P(A)$ be the probability that a student is a philosophy major, and $P(B)$ be the probability that a student is on the swim team. We are given: $P(A) = 0.023$, $P(B) = 0.002$, and $P(A \cup B) = 0.024$.
- Find the intersection probability: We use the Addition Rule of probability: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Substituting the given values: $0.024 = 0.023 + 0.002 - P(A \cap B)$. Solving for $P(A \cap B)$: $P(A \cap B) = 0.025 - 0.024 = 0.001$. This represents the probability that a randomly selected student is both a philosophy major and on the swim team.
- Calculate the conditional probability: We want to find the proportion of the swim team that are philosophy majors, which is the conditional probability $P(A|B)$. The formula is $P(A|B) = \frac{P(A \cap B)}{P(B)}$. Substituting the values: $P(A|B) = \frac{0.001}{0.002} = 0.5$.
- Complete the sentence: The result $0.5$ is equivalent to $50\%$. Therefore, $50\%$ of the members of the swim team are philosophy majors because the probability of being a philosophy major given that the student is on the swim team is $0.5$.