Responder
The sample space includes the following sequences that result in 3 consecutive heads within 6 tosses:
- HHH
- THHH
- HTHH
- HHTH
- HHHH
- TTHHH
- THTHH
- HTHHH
- HHTHH
- HHHTH
- HHHHH
- TTTHHH
- TTHTHH
- TTHHTH
- TTHHHH
- THTHHH
- HTTHHH
- HHTHHH
- HHHTHH
- HHHHTH
- HHHHHH
This sample space is discrete because it consists of a finite number of distinct outcomes.
Solución
To solve the problem, we need to identify the sequences of coin flips that result in 3 heads in succession (HHH) within 6 or fewer tosses.
### Step 1: Identify the Sample Space
We will list all possible sequences of coin flips that result in 3 consecutive heads (HHH) and require 6 or fewer tosses.
1. **3 Tosses**:
- HHH
2. **4 Tosses**:
- THHH
- HTHH
- HHTH
- HHHH (this is also valid as it ends with HHH)
3. **5 Tosses**:
- TTHHH
- THTHH
- THTHH
- HTHHH
- HHTHH
- HHHTH
- HHHHH (this is also valid as it ends with HHH)
4. **6 Tosses**:
- TTTHHH
- TTHTHH
- TTHHTH
- TTHHHH
- THTHHH
- HTTHHH
- HHTHHH
- HHHTHH
- HHHHTH
- HHHHHH (this is also valid as it ends with HHH)
### Step 2: List the Valid Sequences
Now, let's compile the valid sequences:
- **3 Tosses**:
- HHH
- **4 Tosses**:
- THHH
- HTHH
- HHTH
- HHHH
- **5 Tosses**:
- TTHHH
- THTHH
- HTHHH
- HHTHH
- HHHTH
- HHHHH
- **6 Tosses**:
- TTTHHH
- TTHTHH
- TTHHTH
- TTHHHH
- THTHHH
- HTTHHH
- HHTHHH
- HHHTHH
- HHHHTH
- HHHHHH
### Step 3: Determine if the Sample Space is Discrete
A sample space is considered discrete if it consists of a countable number of distinct outcomes. In this case, we have listed a finite number of sequences that can occur when flipping the coin until we get 3 heads in succession.
Since we can count the number of outcomes (which is finite), the sample space is indeed discrete.
### Conclusion
The elements of the sample space that require 6 or fewer tosses are:
- HHH
- THHH
- HTHH
- HHTH
- HHHH
- TTHHH
- THTHH
- HTHHH
- HHTHH
- HHHTH
- HHHHH
- TTTHHH
- TTHTHH
- TTHHTH
- TTHHHH
- THTHHH
- HTTHHH
- HHTHHH
- HHHTHH
- HHHHTH
- HHHHHH
This sample space is discrete because it consists of a finite number of distinct outcomes.
Revisado y aprobado por el equipo de tutoría de UpStudy
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