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

**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, and 8.

So there are 5 favorable outcomes for event A.

The total number of outcomes on the spinner is 8.

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

**Event B:** Rolling a number greater than 3 on the number cube.

The numbers on the number cube are 1, 2, 3, 4, 5, 6. The numbers greater than 3 are 4, 5, and 6.

So there are 3 favorable outcomes for event B.

The total number of outcomes on the number cube is 6.

\[
P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total outcomes on cube}} = \frac{3}{6} = \frac{1}{2}
\]

**Calculating \( P(A \cap B) \):**

Since events A and B are independent (the outcome of the spinner does not affect the outcome of the number cube and vice versa), we can find the joint probability by multiplying their probabilities:

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

Substituting in the values:

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

Thus, the final answer is:

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