Question

Consider a spinner with numbers 1–8 and a number cube with numbers 1–6. Let event A represent landing on a number greater than 3 on the spinner and event B represent rolling a number greater than 3 on the number cube. Find P(A∩B) . Enter your response as a fraction

Answer 1

To find \( P(A \cap B) \), we need to calculate the probabilities of events \( A \) and \( B \) and then their intersection.

1. **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:
\[
P(A) = \frac{\text{Number of outcomes greater than 3}}{\text{Total outcomes}} = \frac{5}{8}
\]

2. **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:
\[
P(B) = \frac{\text{Number of outcomes greater than 3}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2}
\]

3. **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:
\[
\boxed{\frac{5}{16}}
\]