IDNLearn.com offers a comprehensive solution for finding accurate answers quickly. Our platform is designed to provide quick and accurate answers to any questions you may have.
Sagot :
Sure, let's solve this step-by-step to determine which sequence of coin flips is closest to the theoretical probability of 2 heads in every 3 flips, which translates to [tex]\(\frac{2}{3}\)[/tex] or approximately 0.6667 heads.
First, we need to calculate the proportion of heads in each of the sequences:
### Sequence A:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & H & T & T & T & H & T & T & T & T & H & T & T \\ \hline \end{array} \][/tex]
Number of heads (H) = 3
Total number of flips = 12
Proportion of heads for sequence A:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{3}{12} = 0.25 \][/tex]
### Sequence B:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & H & H & H & T & T & H & T & T & H & H & H & H \\ \hline \end{array} \][/tex]
Number of heads (H) = 8
Total number of flips = 12
Proportion of heads for sequence B:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{8}{12} = 0.6667 \][/tex]
### Sequence C:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & H & H & T & T & T & T & H & T & H & T & T & T \\ \hline \end{array} \][/tex]
Number of heads (H) = 4
Total number of flips = 12
Proportion of heads for sequence C:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{4}{12} = 0.3333 \][/tex]
### Sequence D:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & T & H & T & H & T & T & T & H & T & T & H & H \\ \hline \end{array} \][/tex]
Number of heads (H) = 5
Total number of flips = 12
Proportion of heads for sequence D:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{5}{12} = 0.4167 \][/tex]
Now, we compare the proportions to the theoretical proportion of 0.6667 and find the sequence closest to it:
- Difference for sequence A: [tex]\( |0.25 - 0.6667| = 0.4167 \)[/tex]
- Difference for sequence B: [tex]\( |0.6667 - 0.6667| = 0.0 \)[/tex]
- Difference for sequence C: [tex]\( |0.3333 - 0.6667| = 0.3333 \)[/tex]
- Difference for sequence D: [tex]\( |0.4167 - 0.6667| = 0.25 \)[/tex]
The smallest difference is 0.0 for sequence B. Therefore, the sequence that is most consistent with the theoretical model is:
[tex]\[ \text{Sequence B} \][/tex]
First, we need to calculate the proportion of heads in each of the sequences:
### Sequence A:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & H & T & T & T & H & T & T & T & T & H & T & T \\ \hline \end{array} \][/tex]
Number of heads (H) = 3
Total number of flips = 12
Proportion of heads for sequence A:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{3}{12} = 0.25 \][/tex]
### Sequence B:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & H & H & H & T & T & H & T & T & H & H & H & H \\ \hline \end{array} \][/tex]
Number of heads (H) = 8
Total number of flips = 12
Proportion of heads for sequence B:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{8}{12} = 0.6667 \][/tex]
### Sequence C:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & H & H & T & T & T & T & H & T & H & T & T & T \\ \hline \end{array} \][/tex]
Number of heads (H) = 4
Total number of flips = 12
Proportion of heads for sequence C:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{4}{12} = 0.3333 \][/tex]
### Sequence D:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Flip} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text{Result} & T & H & T & H & T & T & T & H & T & T & H & H \\ \hline \end{array} \][/tex]
Number of heads (H) = 5
Total number of flips = 12
Proportion of heads for sequence D:
[tex]\[ \frac{\text{Number of heads}}{\text{Total number of flips}} = \frac{5}{12} = 0.4167 \][/tex]
Now, we compare the proportions to the theoretical proportion of 0.6667 and find the sequence closest to it:
- Difference for sequence A: [tex]\( |0.25 - 0.6667| = 0.4167 \)[/tex]
- Difference for sequence B: [tex]\( |0.6667 - 0.6667| = 0.0 \)[/tex]
- Difference for sequence C: [tex]\( |0.3333 - 0.6667| = 0.3333 \)[/tex]
- Difference for sequence D: [tex]\( |0.4167 - 0.6667| = 0.25 \)[/tex]
The smallest difference is 0.0 for sequence B. Therefore, the sequence that is most consistent with the theoretical model is:
[tex]\[ \text{Sequence B} \][/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.
