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)
.(7 points)
Responses

14
1 fourth

516
5 over 16

18
1 eighth

98

Answer 1

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

**Event A** (landing on a number greater than 3 on the spinner):
- The spinner has numbers from 1 to 8. The numbers greater than 3 are 4, 5, 6, 7, and 8.
- There are 5 favorable outcomes (4, 5, 6, 7, 8) out of 8 possible outcomes.
- Therefore, \( P(A) = \frac{5}{8} \).

**Event B** (rolling a number greater than 3 on the number cube):
- The number cube has numbers from 1 to 6. The numbers greater than 3 are 4, 5, and 6.
- There are 3 favorable outcomes (4, 5, 6) out of 6 possible outcomes.
- Therefore, \( P(B) = \frac{3}{6} = \frac{1}{2} \).

**Finding \( P(A \cap B) \)**:
Since the spinner and the number cube operate independently, we can calculate the combined probability of both events occurring using the formula for the probability of independent events:

\[
P(A \cap B) = P(A) \times P(B)
\]

Calculating it:

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

Thus, the answer is:

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