Question
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.
Ask by Huff Lynch. in Puerto Rico
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The probability that both events \( A \) and \( B \) occur is \( \frac{5}{54} \).
Solution
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:** \(\frac{5}{54}\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find the probability that both independent events \( A \) and \( B \) occur, you multiply their probabilities together. The probability of event \( A \) occurring is \( \frac{5}{6} \) and the probability of event \( B \) occurring is \( \frac{1}{9} \). So, the combined probability \( P(A \cap B) = P(A) \times P(B) \): \[ P(A \cap B) = \frac{5}{6} \times \frac{1}{9} = \frac{5 \times 1}{6 \times 9} = \frac{5}{54} \] Thus, the probability that both events \( A \) and \( B \) occur is \( \frac{5}{54} \).
