To find \( P(A \cap B) \), we first need to determine the probabilities of events \( A \) and \( B \).
Event A: Landing on a number greater than 3 on the spinner.
The spinner has numbers 1 through 8. The numbers greater than 3 are 4, 5, 6, 7, and 8. This gives us:
- Favorable outcomes for \( A \): 5 outcomes (4, 5, 6, 7, 8)
- Total outcomes for the spinner: 8 outcomes
So, \[ P(A) = \frac{5}{8} \]
Event B: Rolling a number greater than 3 on the number cube.
The number cube has numbers 1 through 6. The numbers greater than 3 are 4, 5, and 6. This gives us:
- Favorable outcomes for \( B \): 3 outcomes (4, 5, 6)
- Total outcomes for the number cube: 6 outcomes
So, \[ P(B) = \frac{3}{6} = \frac{1}{2} \]
Since events \( A \) and \( B \) are independent (the outcome of the spinner does not affect the outcome of the number cube and vice versa), we can find \( P(A \cap B) \) using the rule for independent events:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Substituting in the values we found:
\[ P(A \cap B) = \frac{5}{8} \cdot \frac{1}{2} = \frac{5}{16} \]
Thus, the probability \( P(A \cap B) \) is
\[ \boxed{\frac{5}{16}} \]