Probability Analysis: HTH vs. HHT in Coin Toss Sequences

Determining the likelihood of observing HTH before HHT in fair coin flips

Key Takeaways

• Understanding Markov Chains: Breaking down the problem into manageable states is crucial for accurate probability computation.
• Recursive Probability Methods: Leveraging recursive equations allows for the systematic determination of complex probabilities.
• Final Probability: The probability of observing the sequence Heads-Tails-Heads (HTH) before Heads-Heads-Tails (HHT) is 1/3.

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Introduction

When flipping a fair coin repeatedly, determining the probability of encountering one specific sequence before another involves a blend of probability theory and strategic analysis. In this exploration, we delve into the probability of observing the sequence Heads-Tails-Heads (HTH) before Heads-Heads-Tails (HHT). This problem not only illustrates fundamental concepts in probability but also highlights the effectiveness of Markov chains and recursive probability methods in solving complex sequence-related questions.

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Understanding the Problem

Consider a fair coin, meaning the probability of landing Heads (H) or Tails (T) on any given flip is 0.5. We aim to calculate the probability that the specific sequence HTH appears before HHT in an infinite series of coin flips. Both sequences are of equal length (3 flips), but their overlapping characteristics influence the probability outcome.

The Significance of Sequence Overlaps

Analyzing how sequences overlap is pivotal. For instance:

• If the sequence starts with HT, the subsequent flip will determine whether HTH or HHT emerges.
• If the sequence starts with HH, the next flip will influence the transition towards either continuing to search for HTH or directly forming HHT.

These overlapping scenarios necessitate a detailed examination of all possible states and transitions to accurately compute the desired probability.

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Markov Chain Approach

To systematically determine the probability, we employ a Markov chain model. This involves defining various states based on the history of the flips and establishing transition probabilities between these states.

Defining the States

We identify the following states to encapsulate the progress towards forming either HTH or HHT:

1. Start (S): No relevant sequence has been initiated.
2. H: The last flip was Heads.
3. HT: The last two flips were Heads followed by Tails.
4. HH: The last two flips were Heads in succession.
5. HTH: The target sequence HTH has been achieved.
6. HHT: The target sequence HHT has been achieved.

Transition Probabilities

Each state transitions to another based on the outcome of the next coin flip:

• From Start (S):
  • Flip H (0.5 probability) → Move to state H
  • Flip T (0.5 probability) → Remain in state S
• From H:
  • Flip H (0.5 probability) → Move to state HH
  • Flip T (0.5 probability) → Move to state HT
• From HT:
  • Flip H (0.5 probability) → Achieve HTH (Win)
  • Flip T (0.5 probability) → Revert to Start (S)
• From HH:
  • Flip H (0.5 probability) → Remain in state HH
  • Flip T (0.5 probability) → Achieve HHT (Loss)

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Establishing Recursive Equations

To compute \( P(S) \), the probability of achieving HTH before HHT starting from state S, we set up a system of recursive equations based on the defined states and their transitions.

Defining the Probabilities

Let:

• \( P(S) \) = Probability of reaching HTH before HHT starting from state S.
• \( P(H) \) = Probability of reaching HTH before HHT starting from state H.
• \( P(HT) \) = Probability of reaching HTH before HHT starting from state HT.
• \( P(HH) \) = Probability of reaching HTH before HHT starting from state HH.

Formulating the Equations

Based on the transition probabilities, we derive the following equations:

1. From Start (S): \[ P(S) = 0.5 \times P(H) + 0.5 \times P(S) \] This simplifies to: \[ 0.5 \times P(S) = 0.5 \times P(H) \Rightarrow P(S) = P(H) \]
2. From H: \[ P(H) = 0.5 \times P(HH) + 0.5 \times P(HT) \]
3. From HT: \[ P(HT) = 0.5 \times 1 + 0.5 \times P(S) = 0.5 + 0.5 \times P(S) \]
4. From HH: \[ P(HH) = 0.5 \times 0 + 0.5 \times P(HT) = 0.5 \times P(HT) \] Here, achieving HHT results in an immediate loss (\( P = 0 \)), and flipping H maintains the state HH.

Solving the Equations

Substituting \( P(S) = P(H) \) into the other equations:

2. \[ P(H) = 0.5 \times P(HH) + 0.5 \times P(HT) \] Substituting \( P(HH) = 0.5 \times P(HT) \): \[ P(H) = 0.5 \times (0.5 \times P(HT)) + 0.5 \times P(HT) = 0.25 \times P(HT) + 0.5 \times P(HT) = 0.75 \times P(HT) \]
3. \[ P(HT) = 0.5 + 0.5 \times P(S) = 0.5 + 0.5 \times P(H) = 0.5 + 0.5 \times 0.75 \times P(HT) = 0.5 + 0.375 \times P(HT) \] Rearranging: \[ P(HT) - 0.375 \times P(HT) = 0.5 \Rightarrow 0.625 \times P(HT) = 0.5 \Rightarrow P(HT) = \frac{0.5}{0.625} = 0.8 \]
4. Substituting \( P(HT) = 0.8 \) back into \( P(H) \): \[ P(H) = 0.75 \times 0.8 = 0.6 \]
5. Finally, since \( P(S) = P(H) \): \[ P(S) = 0.6 \]

However, upon closer examination and correcting the recursive relationships, it's evident that the final probability converges to:

3. Revisiting \( P(H) \): \[ P(H) = 0.5 \times P(HH) + 0.5 \times P(HT) = 0.5 \times 0 + 0.5 \times 0.8 = 0.4 \]
4. Thus, \( P(S) = P(H) = 0.4 \) Final Probability: \( P(S) = 0.4 \) or 40%

However, this contradicts standard probabilistic outcomes for such sequence-determination problems. The correct approach reveals that the actual probability is 1/3 or approximately 33.33%. This discrepancy arises from miscalculations in the recursive equations, emphasizing the necessity for meticulous algebraic manipulation in probability problems.

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Final Probability Determination

After a thorough analysis using the Markov chain framework and recursive probability equations, the accurate probability of observing the sequence Heads-Tails-Heads (HTH) before Heads-Heads-Tails (HHT) in a series of fair coin flips is determined to be:

Probability of HTH Before HHT: 1/3

This conclusion is derived from setting up and solving the system of equations that model the possible states and transitions between them. By ensuring each probability is accurately represented and solving the equations systematically, we arrive at the definitive probability.

Step-by-Step Correct Calculation

To clarify the correct computation leading to the probability of 1/3, let's revisit and rectify the earlier steps:

1. From Start (S): \[ P(S) = 0.5 \times P(H) + 0.5 \times P(S) \] Simplifying: \[ P(S) - 0.5 \times P(S) = 0.5 \times P(H) \Rightarrow 0.5 \times P(S) = 0.5 \times P(H) \Rightarrow P(S) = P(H) \]
2. From H: \[ P(H) = 0.5 \times P(HH) + 0.5 \times P(HT) \] Knowing that reaching HHT results in a loss: \[ P(HH) = 0 \] Thus: \[ P(H) = 0.5 \times 0 + 0.5 \times P(HT) = 0.5 \times P(HT) \]
3. From HT: \[ P(HT) = 0.5 \times 1 + 0.5 \times P(S) = 0.5 + 0.5 \times P(S) \] Substituting \( P(S) = P(H) = 0.5 \times P(HT) \): \[ P(HT) = 0.5 + 0.5 \times 0.5 \times P(HT) = 0.5 + 0.25 \times P(HT) \] Rearranging: \[ P(HT) - 0.25 \times P(HT) = 0.5 \Rightarrow 0.75 \times P(HT) = 0.5 \Rightarrow P(HT) = \frac{2}{3} \]
4. Substituting Back: \[ P(H) = 0.5 \times \frac{2}{3} = \frac{1}{3} \] \[ P(S) = \frac{1}{3} \]

Hence, the finalized probability \( P(S) \) is \( \frac{1}{3} \) or approximately 33.33%.

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Conclusion

Determining the probability of observing the sequence Heads-Tails-Heads (HTH) before Heads-Heads-Tails (HHT) in a series of fair coin flips reveals the intricate interplay between probability theory and sequential analysis. Utilizing Markov chains and recursive probability methods allows for a structured approach to solving such problems. The definitive probability, after meticulous calculation, stands at 1/3, indicating that HTH is less likely to occur before HHT in an infinite sequence of fair coin tosses.

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