Question

There is a spinner with 9 equal areas, numbered 1 through 9. If the spinner is spun one time, what is the probability that the result is a multiple of 3 or a multiple of 2?(1 point)
Responses

2/3
2 thirds

4/9
4 ninths

7/9
7 ninths

1/3
1 third

Answer 1

To find the probability that the result of spinning the spinner is a multiple of 3 or a multiple of 2, we first identify the numbers on the spinner that meet these conditions.

The numbers on the spinner are 1, 2, 3, 4, 5, 6, 7, 8, and 9.

**Multiples of 3** from 1 to 9 are:
- 3, 6, 9

So, there are 3 multiples of 3.

**Multiples of 2** from 1 to 9 are:
- 2, 4, 6, 8

So, there are 4 multiples of 2.

Now, we need to determine if there's any overlap (i.e., numbers that are both multiples of 2 and multiples of 3). The only common number is:
- 6

This means there are 3 multiples of 3, 4 multiples of 2, and 1 number that is counted in both categories.

Using the principle of inclusion-exclusion, we calculate the total number of successful outcomes:
\[
\text{Total} = \text{(Multiples of 3)} + \text{(Multiples of 2)} - \text{(Overlap)}
\]
\[
\text{Total} = 3 + 4 - 1 = 6
\]

There are 6 favorable outcomes (the numbers 2, 3, 4, 6, 8, 9).

The total number of outcomes is 9 (since there are 9 sections in the spinner).

Thus, the probability \( P \) of landing on a number that is a multiple of 3 or a multiple of 2 is:
\[
P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{6}{9} = \frac{2}{3}
\]

Therefore, the answer is:
**2/3 (2 thirds)**.