A telephone sales representative makes successive calls to potential customers, and the result of each call is recorded as a sale (S) or a no sale (N). Calls are continued until either two sales are made (these sales need not be consecutive), two consecutive calls result in no sale, or a total of 4 calls is made. What is the sample space for this situation?

To determine the sample space for this situation, we need to list all possible sequences of outcomes based on the given stopping conditions:

### Stopping Conditions:
1. **Two sales (S)**: The process stops as soon as two sales are made.
2. **Two consecutive no sales (NN)**: The process stops if two consecutive calls result in no sale.
3. **Four calls**: If neither condition 1 nor 2 occurs after four calls, the process automatically stops.

Each call can result in either a sale (S) or no sale (N), and the process may terminate after 2, 3, or 4 calls based on the conditions above.

### Constructing the Sample Space
We will consider all possible outcomes that meet these conditions:

1. **If two sales (S) occur before the fourth call**:
   - SS
   - SNS
   - NSS
   - SNSN (4 calls, with only two sales)

2. **If two consecutive no sales (NN) occur**:
   - NN
   - SNN
   - NSNN

3. **If none of the above stopping conditions occur, the sequence must continue to four calls**:
   - SSSS (all calls are sales)
   - SSSN
   - SSNS
   - SSNN
   - SNSS
   - SNSN
   - SNSN (repeated from above)
   - SNNS
   - SNSS

### Final Sample Space
So, the complete sample space, listing all possible stopping conditions, is:

\[
\{SS, SNS, NSS, SNSN, NN, SNN, NSNN, SSSS, SSSN, SSNS, SSNN, SNSS, SNNS\}
\]

### Explanation:
- The sample space includes all possible sequences where the process could end based on the stopping rules.
- It is important to note that the maximum number of calls is 4, and the process may stop sooner based on achieving either two sales or two consecutive no sales.

Would you like a more detailed breakdown or any specific clarifications?

Here are 5 related questions:
1. What is the probability of the process stopping after exactly 2 calls?
2. What if the condition for stopping was 3 consecutive no sales instead of 2?
3. How would you model this situation using a decision tree?
4. What is the total number of sequences possible without the stopping conditions?
5. How could you calculate the expected number of calls until the process stops?

**Tip:** When working with sample spaces, always check the conditions carefully, as they limit the possible outcomes significantly.

Explore the sample space for a scenario where a telephone sales representative makes calls to potential customers, recording results as sales or no sales. This problem involves combinatorics and probability concepts, examining stopping conditions after two sales or two consecutive no sales. Suitable for students in Grades 9-12.

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