To determine which sequence of coin flips is the most consistent with an unbiased coin, we need to count the number of heads (H) and tails (T) in each sequence and then evaluate the difference between the number of heads and tails. The most consistent sequence will have the smallest difference between the number of heads and tails.
Let's analyze each sequence:
Sequence 1:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Flip & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Result & T & H & T & H & T & T & T & H & T & T & H & H \\
\hline
\end{array}
\][/tex]
Number of heads [tex]\( (H) \)[/tex] = 5, Number of tails [tex]\( (T) \)[/tex] = 7
Difference = [tex]\( |5 - 7| = 2 \)[/tex]
Sequence 2:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Flip & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Result & H & T & T & T & H & T & T & T & T & H & T & T \\
\hline
\end{array}
\][/tex]
Number of heads [tex]\( (H) \)[/tex] = 3, Number of tails [tex]\( (T) \)[/tex] = 9
Difference = [tex]\( |3 - 9| = 6 \)[/tex]
Sequence 3:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Flip & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Result & H & H & T & T & T & T & H & T & H & T & T & T \\
\hline
\end{array}
\][/tex]
Number of heads [tex]\( (H) \)[/tex] = 5, Number of tails [tex]\( (T) \)[/tex] = 7
Difference = [tex]\( |5 - 7| = 2 \)[/tex]
Sequence 4:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Flip & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Result & H & H & H & T & T & H & T & T & H & H & H & H \\
\hline
\end{array}
\][/tex]
Number of heads [tex]\( (H) \)[/tex] = 8, Number of tails [tex]\( (T) \)[/tex] = 4
Difference = [tex]\( |8 - 4| = 4 \)[/tex]
Comparing the differences, we have:
- Sequence 1: Difference = 2
- Sequence 2: Difference = 6
- Sequence 3: Difference = 2
- Sequence 4: Difference = 4
The sequences with the smallest difference (Sequence 1 and Sequence 3) are the most consistent with an unbiased coin.
However, since we need to pick the first sequence that meets this criterion, the correct answer is:
Sequence 1
