"Statistics". Acoin is flipped until 3 heads in succession occur. List only those elements of the sample space that require 6 or less tosses. Is this a discrete sample space? Explain.

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.