Certainly! Let's solve this question step-by-step to find which of the given sequences of coin flips is the most probable for a biased coin that lands on heads two-thirds of the time.
1. Understanding the Coin Bias:
- The probability of landing on heads (H) is [tex]\(\frac{2}{3}\)[/tex].
- The probability of landing on tails (T) is [tex]\(1 - \frac{2}{3} = \frac{1}{3}\)[/tex].
2. List of Sequences:
- THTHTTTHTTHH
- HHTTTTHTHTTT
- HHHTTHTTHHHH
- HTTTHTTTTHTT
3. Calculate the Probability of Each Sequence:
For each sequence, we will multiply the probabilities of getting heads or tails for each flip in the sequence.
4. Calculations:
Let's summarize the probabilities (which have been calculated carefully):
- Sequence "THTHTTTHTTHH":
[tex]\[
\text{Probability} = 6.0213645541085506 \times 10^{-5}
\][/tex]
- Sequence "HHTTTTHTHTTT":
[tex]\[
\text{Probability} = 3.0106822770542753 \times 10^{-5}
\][/tex]
- Sequence "HHHTTHTTHHHH":
[tex]\[
\text{Probability} = 0.00048170916432868383
\][/tex]
- Sequence "HTTTHTTTTHTT":
[tex]\[
\text{Probability} = 1.5053411385271375 \times 10^{-5}
\][/tex]
5. Identifying the Most Probable Sequence:
- Among the calculated probabilities, we identify the highest probability.
6. Comparison:
- [tex]\(6.0213645541085506 \times 10^{-5}\)[/tex]
- [tex]\(3.0106822770542753 \times 10^{-5}\)[/tex]
- [tex]\(0.00048170916432868383\)[/tex]
- [tex]\(1.5053411385271375 \times 10^{-5}\)[/tex]
Clearly, [tex]\(0.00048170916432868383\)[/tex] is the largest probability.
7. Conclusion:
- Therefore, the sequence "HHHTTHTTHHHH" is the most probable sequence given the biased coin.
Answer:
The sequence "HHHTTHTTHHHH" is the most probable sequence for the biased coin.
