calculate conditional probability of swim team members being philosophy majors
Check the final answer first, then review the worked steps.
Problem
calculate conditional probability of swim team members being philosophy majors
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 $0.5$.