In an experiment, the probability that event \( A \) occurs is \( \frac{5}{6} \) and the probability that event \( B \) occurs is \( \frac{1}{9} \). 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 fact that if \( A \) and \( B \) are independent, the probability of both events happening together is the product of their individual probabilities. Given: - Probability of \( A \), \( P(A) = \frac{5}{6} \) - Probability of \( B \), \( P(B) = \frac{1}{9} \) The probability that both \( A \) and \( B \) occur is: \[ P(A \text{ and } B) = P(A) \times P(B) = \frac{5}{6} \times \frac{1}{9} = \frac{5}{54} \] The fraction \( \frac{5}{54} \) is already in its simplest form since 5 and 54 have no common divisors other than 1. **Answer:** \(\frac{5}{54}\)