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 \( B \) 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:

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