In an experiment, the probability that event \( A \) occurs is \( \frac{7}{9} \) and the probability that event \( B \) occurs is \( \frac{5}{6} \). If \( A \) and \( B \) are independent events, what is the probability that \( A \) and \( B \) both occur? Simplify any fractions.

To find the probability that both events \( A \) and \( B \) occur, we use the formula for independent events: \[ P(A \cap B) = P(A) \cdot P(B) \] Substituting the probabilities given: \[ P(A) = \frac{7}{9}, \quad P(B) = \frac{5}{6} \] Now, multiply these two probabilities: \[ P(A \cap B) = \frac{7}{9} \cdot \frac{5}{6} = \frac{7 \cdot 5}{9 \cdot 6} = \frac{35}{54} \] The final probability that both events \( A \) and \( B \) occur is: \[ \boxed{\frac{35}{54}} \]