Question

in an experiment, the probability that event a occurs is 1/4, the probability that event b occurs is 3/7, and the probability that events a and b both occur is 2/9. what is the probability that a occurs given that b occurs? simplify any fractions.

Explanation:

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( P(A|B) \) is the probability of \( A \) given \( B \), \( P(A \cap B) \) is the probability of both \( A \) and \( B \) occurring, and \( P(B) \) is the probability of \( B \) occurring.

Step2: Identify Given Values

We know that \( P(A \cap B)=\frac{2}{9} \) and \( P(B)=\frac{3}{7} \).

Step3: Substitute Values into Formula

Substitute these values into the formula: \( P(A|B) = \frac{\frac{2}{9}}{\frac{3}{7}} \).

Step4: Divide the Fractions

To divide fractions, we multiply by the reciprocal: \( \frac{2}{9} \times \frac{7}{3}=\frac{14}{27} \).

Answer:

\(\frac{14}{27}\)