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Part 1
It is equally probable that the pointer on the spinner shown will land on any one of the eight​ regions, numbered 1 through 8. If the pointer lands on a​ borderline, spin again. Find the probability that the pointer will stop on an odd number or a number less than 8.

Answer 1

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:

\[
\frac{7}{8}
\]