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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]
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