Questions: There is a spinner with 8 equal areas, numbered 1 through 8. If the spinner is spun one time, what is the probability that the result is a multiple of 2 or a multiple of 3?

Solution Steps

The spinner has 8 equal areas numbered from 1 to 8. Therefore, the total number of possible outcomes when the spinner is spun once is \( 8 \).

The multiples of 2 between 1 and 8 are \( 2, 4, 6, 8 \). There are \( 4 \) favorable outcomes for multiples of 2.

The multiples of 3 between 1 and 8 are \( 3, 6 \). There are \( 2 \) favorable outcomes for multiples of 3.

The number \( 6 \) is a multiple of both 2 and 3. Therefore, there is \( 1 \) overlapping outcome.

Using the principle of inclusion-exclusion, the total number of favorable outcomes is: \[ \text{Favorable outcomes} = (\text{Multiples of 2}) + (\text{Multiples of 3}) - (\text{Overlapping outcomes}) = 4 + 2 - 1 = 5 \]

The probability \( P \) of the spinner landing on a multiple of 2 or 3 is: \[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{5}{8} \]

Final Answer