To determine \( P(A \cap B) \), we first need to find the probabilities of events \( A \) and \( B \).
Event A: Landing on a number greater than 3 on the spinner.
The numbers on the spinner are 1 through 8. The numbers greater than 3 are 4, 5, 6, 7, and 8. Therefore, there are 5 favorable outcomes.
\[ P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total outcomes on the spinner}} = \frac{5}{8} \]
Event B: Rolling a number greater than 3 on the number cube.
The numbers on the cube are 1 through 6. The numbers greater than 3 are 4, 5, and 6. Therefore, there are 3 favorable outcomes.
\[ P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total outcomes on the number cube}} = \frac{3}{6} = \frac{1}{2} \]
Since the spinner and the number cube operate independently, we can find \( P(A \cap B) \) by multiplying the probabilities of events \( A \) and \( B \):
\[ P(A \cap B) = P(A) \cdot P(B) = \left( \frac{5}{8} \right) \cdot \left( \frac{1}{2} \right) = \frac{5}{16} \]
Thus, the probability \( P(A \cap B) \) is
\[ \boxed{\frac{5}{16}} \]
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