To find the probability that the spinner stops on an odd number or a number less than 8, we can begin by identifying the outcomes that meet either of these conditions.
Step 1: Identify the total outcomes. The spinner has 8 regions numbered 1 through 8. Therefore, the total number of outcomes (when disregarding spins on the borderline) is \( 8 \).
Step 2: Identify the odd numbers. The odd numbers on the spinner are:
- 1
- 3
- 5
- 7
So, we have 4 odd numbers.
Step 3: Identify the numbers less than 8. The numbers less than 8 on the spinner are:
- 1
- 2
- 3
- 4
- 5
- 6
- 7
So, we have 7 numbers that are less than 8.
Step 4: Identify the overlap. The odd numbers that are also less than 8 are:
- 1
- 3
- 5
- 7
We see that all odd numbers (1, 3, 5, 7) are included in the numbers that are less than 8. Thus, the overlap consists of 4 numbers.
Step 5: Apply the formula for the union of two sets. To find the probability of either event occurring (odd numbers or numbers less than 8), we use the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Where:
- \( P(A) \) = Probability of landing on an odd number = \( \frac{4}{8} \)
- \( P(B) \) = Probability of landing on a number less than 8 = \( \frac{7}{8} \)
- \( P(A \cap B) \) = Probability of landing on an odd number that is also less than 8 = \( \frac{4}{8} \)
Step 6: Calculate individual probabilities. Substituting into the formula:
\[ P(A \cup B) = \frac{4}{8} + \frac{7}{8} - \frac{4}{8} \]
This simplifies to:
\[ P(A \cup B) = \frac{7}{8} \]
Step 7: Final result. Thus, the probability that the pointer will stop on an odd number or a number less than 8 is:
\[ \frac{7}{8} \]
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