Question

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.(1 point)

Answer 1

To find the probability \( P(A \cap B) \), we first need to determine the probabilities of events \( A \) and \( B \).

1. **Calculate \( P(A) \)**: Event \( A \) represents landing on a number greater than 3 on the spinner numbered from 1 to 8.
- The numbers greater than 3 on the spinner are: 4, 5, 6, 7, and 8.
- Therefore, there are 5 favorable outcomes.
- The total number of outcomes on the spinner is 8.

\[
P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{8}
\]

2. **Calculate \( P(B) \)**: Event \( B \) represents rolling a number greater than 3 on a number cube (1 to 6).
- The numbers greater than 3 on the cube are: 4, 5, and 6.
- Thus, there are 3 favorable outcomes.
- The total number of outcomes on the number cube is 6.

\[
P(B) = \frac{3}{6} = \frac{1}{2}
\]

3. **Calculate \( P(A \cap B) \)**: Since the outcomes of the spinner and the number cube are independent events, we can find \( P(A \cap B) \) by multiplying the probabilities of \( A \) and \( B \).

\[
P(A \cap B) = P(A) \cdot P(B) = \left(\frac{5}{8}\right) \cdot \left(\frac{1}{2}\right) = \frac{5 \times 1}{8 \times 2} = \frac{5}{16}
\]

Thus, the probability \( P(A \cap B) \) is:

\[
\boxed{\frac{5}{16}}
\]