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]
