Answer :
To determine which sequence of coin flips is most consistent with the theoretical model where the probability of getting heads (H) is [tex]\(\frac{2}{3}\)[/tex] and the probability of getting tails (T) is [tex]\(\frac{1}{3}\)[/tex], we can follow a step-by-step analysis:
### Step 1: Theoretical Probability
The theoretical probability of heads [tex]\( H \)[/tex] is:
[tex]\[ P(H) = \frac{2}{3} \approx 0.6667 \][/tex]
### Step 2: Calculating Empirical Probability of Heads in Each Sequence
#### Sequence A:
[tex]\[ \text{HHTTTHTHTTTT} \][/tex]
Count of Heads (H): 4
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{A}(H) = \frac{4}{12} \approx 0.3333 \][/tex]
#### Sequence B:
[tex]\[ \text{HHHTTHHTHHHH} \][/tex]
Count of Heads (H): 9
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{B}(H) = \frac{9}{12} = 0.75 \][/tex]
#### Sequence C:
[tex]\[ \text{HTTTHHTTTHTT} \][/tex]
Count of Heads (H): 4
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{C}(H) = \frac{4}{12} \approx 0.3333 \][/tex]
#### Sequence D:
[tex]\[ \text{THTHTTHTTTHH} \][/tex]
Count of Heads (H): 5
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{D}(H) = \frac{5}{12} \approx 0.4167 \][/tex]
### Step 3: Calculate the Difference Between Empirical and Theoretical Probabilities
We will compute the absolute difference between the empirical probability of heads and the theoretical probability for each sequence.
#### Sequence A:
[tex]\[ |P_{A}(H) - \frac{2}{3}| = |0.3333 - 0.6667| \approx 0.3333 \][/tex]
#### Sequence B:
[tex]\[ |P_{B}(H) - \frac{2}{3}| = |0.75 - 0.6667| \approx 0.0833 \][/tex]
#### Sequence C:
[tex]\[ |P_{C}(H) - \frac{2}{3}| = |0.3333 - 0.6667| \approx 0.3333 \][/tex]
#### Sequence D:
[tex]\[ |P_{D}(H) - \frac{2}{3}| = |0.4167 - 0.6667| \approx 0.2500 \][/tex]
### Step 4: Identifying the Sequence with Minimum Difference
The sequence that is most consistent with the theoretical model will have the smallest difference:
- Sequence A: [tex]\(\approx 0.3333\)[/tex]
- Sequence B: [tex]\(\approx 0.0833\)[/tex]
- Sequence C: [tex]\(\approx 0.3333\)[/tex]
- Sequence D: [tex]\(\approx 0.2500\)[/tex]
Among these, Sequence B has the minimum difference of [tex]\(\approx 0.0833\)[/tex].
### Conclusion
The sequence most consistent with the theoretical model is:
[tex]\[ \boxed{B} \][/tex]
### Step 1: Theoretical Probability
The theoretical probability of heads [tex]\( H \)[/tex] is:
[tex]\[ P(H) = \frac{2}{3} \approx 0.6667 \][/tex]
### Step 2: Calculating Empirical Probability of Heads in Each Sequence
#### Sequence A:
[tex]\[ \text{HHTTTHTHTTTT} \][/tex]
Count of Heads (H): 4
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{A}(H) = \frac{4}{12} \approx 0.3333 \][/tex]
#### Sequence B:
[tex]\[ \text{HHHTTHHTHHHH} \][/tex]
Count of Heads (H): 9
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{B}(H) = \frac{9}{12} = 0.75 \][/tex]
#### Sequence C:
[tex]\[ \text{HTTTHHTTTHTT} \][/tex]
Count of Heads (H): 4
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{C}(H) = \frac{4}{12} \approx 0.3333 \][/tex]
#### Sequence D:
[tex]\[ \text{THTHTTHTTTHH} \][/tex]
Count of Heads (H): 5
Total flips: 12
Empirical Probability of Heads:
[tex]\[ P_{D}(H) = \frac{5}{12} \approx 0.4167 \][/tex]
### Step 3: Calculate the Difference Between Empirical and Theoretical Probabilities
We will compute the absolute difference between the empirical probability of heads and the theoretical probability for each sequence.
#### Sequence A:
[tex]\[ |P_{A}(H) - \frac{2}{3}| = |0.3333 - 0.6667| \approx 0.3333 \][/tex]
#### Sequence B:
[tex]\[ |P_{B}(H) - \frac{2}{3}| = |0.75 - 0.6667| \approx 0.0833 \][/tex]
#### Sequence C:
[tex]\[ |P_{C}(H) - \frac{2}{3}| = |0.3333 - 0.6667| \approx 0.3333 \][/tex]
#### Sequence D:
[tex]\[ |P_{D}(H) - \frac{2}{3}| = |0.4167 - 0.6667| \approx 0.2500 \][/tex]
### Step 4: Identifying the Sequence with Minimum Difference
The sequence that is most consistent with the theoretical model will have the smallest difference:
- Sequence A: [tex]\(\approx 0.3333\)[/tex]
- Sequence B: [tex]\(\approx 0.0833\)[/tex]
- Sequence C: [tex]\(\approx 0.3333\)[/tex]
- Sequence D: [tex]\(\approx 0.2500\)[/tex]
Among these, Sequence B has the minimum difference of [tex]\(\approx 0.0833\)[/tex].
### Conclusion
The sequence most consistent with the theoretical model is:
[tex]\[ \boxed{B} \][/tex]
