Pregunta
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"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.

Ask by Santiago Griffin. in Pakistan
Jan 22,2025

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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.

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Mind Expander

In this scenario, the possible sequences that could lead to three consecutive heads (HHH) within six tosses include HHH, THH, HTH, HHT, TTHH, HTTH, HHTT, THTHH, HTTTH, and more. However, the sequences that are valid must terminate after reaching HHH, specifically: HHH, THH, HTH, HHT, TTHH, and HTTH. Yes, this is a discrete sample space because it consists of a finite number of distinct outcomes defined by the condition of obtaining three consecutive heads. Each flip results in a well-defined outcome, and all elements can be clearly counted, making it discrete!

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