Question
Students are taking two tests. The probability of passing both tests is 0.6. The probability of passing the first test is 0.8. What is the probability of passing the second test given that the class passed the first test?
Answers
GPT-4o mini
Let \( A \) be the event of passing the first test and \( B \) be the event of passing the second test. We are given that:
- \( P(A) = 0.8 \) (probability of passing the first test),
- \( P(A \cap B) = 0.6 \) (probability of passing both tests).
We are looking for the probability of passing the second test given that the first test was passed, which is \( P(B | A) \).
Using the definition of conditional probability, we have:
\[
P(B | A) = \frac{P(A \cap B)}{P(A)}
\]
Substituting the known values:
\[
P(B | A) = \frac{P(A \cap B)}{P(A)} = \frac{0.6}{0.8}
\]
Calculating this gives:
\[
P(B | A) = \frac{0.6}{0.8} = 0.75
\]
Therefore, the probability of passing the second test given that the first test was passed is \( \boxed{0.75} \).
- \( P(A) = 0.8 \) (probability of passing the first test),
- \( P(A \cap B) = 0.6 \) (probability of passing both tests).
We are looking for the probability of passing the second test given that the first test was passed, which is \( P(B | A) \).
Using the definition of conditional probability, we have:
\[
P(B | A) = \frac{P(A \cap B)}{P(A)}
\]
Substituting the known values:
\[
P(B | A) = \frac{P(A \cap B)}{P(A)} = \frac{0.6}{0.8}
\]
Calculating this gives:
\[
P(B | A) = \frac{0.6}{0.8} = 0.75
\]
Therefore, the probability of passing the second test given that the first test was passed is \( \boxed{0.75} \).