Question

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

Explanation:

Step1: Recall the formula for conditional probability

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: Rearrange the formula to solve for \( P(A \cap B) \)

Multiply both sides of the formula by \( P(B) \) to get \( P(A \cap B) = P(A|B) \times P(B) \).

Step3: Substitute the given values

We know that \( P(B) = \frac{2}{9} \) and \( P(A|B) = \frac{4}{7} \). Substituting these values into the formula, we get \( P(A \cap B) = \frac{4}{7} \times \frac{2}{9} \).

Step4: Multiply the fractions

To multiply the fractions, multiply the numerators together and the denominators together: \( \frac{4 \times 2}{7 \times 9} = \frac{8}{63} \).

Answer:

\(\frac{8}{63}\)