Question
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.
Ask by Christensen Vargas. in Puerto Rico
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The probability that both \( A \) and \( B \) occur is \( \frac{35}{54} \).
Solution
To find the probability that both events \( A \) and \( B \) occur when they are independent, you can multiply their individual probabilities.
Given:
- \( P(A) = \frac{7}{9} \)
- \( P(B) = \frac{5}{6} \)
Since \( A \) and \( B \) are independent:
\[
P(A \text{ and } B) = P(A) \times P(B) = \frac{7}{9} \times \frac{5}{6} = \frac{35}{54}
\]
So, the probability that both \( A \) and \( B \) occur is:
\[
\boxed{\dfrac{35}{54}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
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: \[ \boxed{\frac{35}{54}} \]
