To find \( P(A \cap B) \), we need to calculate the probabilities of events \( A \) and \( B \) and then their intersection.
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Calculate \( P(A) \): This is the probability of landing on a number greater than 3 on the spinner.
- The spinner has numbers 1 through 8, which are: 1, 2, 3, 4, 5, 6, 7, 8.
- The numbers greater than 3 are: 4, 5, 6, 7, 8 (which are 5 outcomes).
- Therefore, the probability \( P(A) \) is: \[ P(A) = \frac{\text{Number of outcomes greater than 3}}{\text{Total outcomes}} = \frac{5}{8} \]
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Calculate \( P(B) \): This is the probability of rolling a number greater than 3 on the number cube.
- The number cube has numbers 1 through 6, which are: 1, 2, 3, 4, 5, 6.
- The numbers greater than 3 are: 4, 5, 6 (which are 3 outcomes).
- Therefore, the probability \( P(B) \) is: \[ P(B) = \frac{\text{Number of outcomes greater than 3}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]
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Calculate \( P(A \cap B) \): Since the spinner and the number cube are independent events, the probability of both events occurring together is: \[ P(A \cap B) = P(A) \times P(B) \] Substituting the probabilities: \[ P(A \cap B) = \frac{5}{8} \times \frac{1}{2} = \frac{5 \times 1}{8 \times 2} = \frac{5}{16} \]
Thus, the final answer is: \[ \boxed{\frac{5}{16}} \]