To find \( P(A \cap B) \), we need to calculate the probabilities of events \( A \) and \( B \) separately and then use the fact that these two events are independent.
Step 1: Calculate \( P(A) \), the probability of landing on a number greater than 3 on the spinner.
The spinner has the numbers 1 through 8. The numbers greater than 3 are 4, 5, 6, 7, and 8.
- There are 5 favorable outcomes (4, 5, 6, 7, 8).
- The total number of outcomes on the spinner is 8.
Thus, the probability is:
\[ P(A) = \frac{5}{8} \]
Step 2: Calculate \( P(B) \), the probability of rolling a number greater than 3 on the number cube.
The number cube has the numbers 1 through 6. The numbers greater than 3 are 4, 5, and 6.
- There are 3 favorable outcomes (4, 5, 6).
- The total number of outcomes on the cube is 6.
Thus, the probability is:
\[ P(B) = \frac{3}{6} = \frac{1}{2} \]
Step 3: Calculate \( P(A \cap B) \).
Since events \( A \) and \( B \) are independent, we can use the multiplication rule:
\[ P(A \cap B) = P(A) \times P(B) \]
Substituting the values we calculated:
\[ P(A \cap B) = \frac{5}{8} \times \frac{1}{2} = \frac{5 \cdot 1}{8 \cdot 2} = \frac{5}{16} \]
Therefore, the final answer is:
\[ \boxed{\frac{5}{16}} \]
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