Question

Question
Lucia has a spinner with the numbers 1-8 on it and a number cube with the 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. See the pictures below as an example. Make sure to enter your answer as a fraction.



(1 point)


What is P(A∩B) =

Answer 1

To calculate \( P(A \cap B) \), we need to determine the probabilities of events A and B occurring simultaneously.

1. **Calculate the probability of event A** (landing on a number greater than 3 on the spinner):
- The numbers on the spinner are 1, 2, 3, 4, 5, 6, 7, 8.
- The numbers greater than 3 are: 4, 5, 6, 7, 8 (which are 5 outcomes).
- There are 8 total outcomes on the spinner.
- Thus, the probability of event A is:
\[
P(A) = \frac{5}{8}
\]

2. **Calculate the probability of event B** (rolling a number greater than 3 on the number cube):
- The numbers on the cube are 1, 2, 3, 4, 5, 6.
- The numbers greater than 3 are: 4, 5, 6 (which are 3 outcomes).
- There are 6 total outcomes on the cube.
- Thus, the probability of event B is:
\[
P(B) = \frac{3}{6} = \frac{1}{2}
\]

3. **Calculate the probability of both events A and B occurring simultaneously (\( P(A \cap B) \))**:
- Since the spinner and the cube are independent of each other, we can calculate the joint probability by multiplying the probabilities:
\[
P(A \cap B) = P(A) \times P(B) = \frac{5}{8} \times \frac{1}{2}
\]
- Now performing the multiplication:
\[
P(A \cap B) = \frac{5 \times 1}{8 \times 2} = \frac{5}{16}
\]

Thus, the probability \( P(A \cap B) = \frac{5}{16} \).