Coin Sequence Probability: HHT vs. HTT

Solution & Explanation

To solve this problem, we need to determine which sequence, HHT (Heads-Heads-Tails) or HTT (Heads-Tails-Tails), is more likely to appear first when flipping a fair coin repeatedly. We'll approach this by analyzing the probability of each sequence occurring first.

Step-by-Step Analysis:

1. Initial Conditions:

   • We start flipping a fair coin, which means each flip has an equal probability of being either Heads (H) or Tails (T), specifically 0.5 for each.
   • We're interested in the sequences HHT and HTT.

2. Breaking Down the Sequences:

   • Both sequences begin with an H, so the first flip must be H for either sequence to start.
   • After an initial H, the sequences diverge based on the next flips.

3. Probability of HHT appearing first:

   • If the first flip is H, the second flip needs to be H again for HHT to continue.
   • The probability of getting two consecutive H's (HH) is 0.5 * 0.5 = 0.25.
   • After obtaining HH, the next flip must be T for HHT to complete, with a probability of 0.5.
   • Therefore, the probability of HHT occurring after the first H is 0.25 * 0.5 = 0.125.

4. Probability of HTT appearing first:

   • If the first flip is H, the second flip must be T for HTT to continue.
   • The probability of HT occurring is 0.5 * 0.5 = 0.25.
   • After obtaining HT, the next flip must be T for HTT to complete, with a probability of 0.5.
   • Therefore, the probability of HTT occurring after the first H is 0.25 * 0.5 = 0.125.

5. Comparing the Probabilities:

   • The analysis above seems to suggest that both sequences have the same probability of occurring after the first H, but this isn't the complete picture.
   • The key difference arises when considering the sequences over multiple flips, especially when neither sequence completes after initial attempts.

6. Recursive Probability Analysis:

   • If after HT, another H appears, it resets the sequence for HHT, increasing its likelihood.
   • Conversely, HTT requires two consecutive T's after H, which is less likely compared to a single T needed for HHT.
   • This recursive nature gives HHT an edge in probability.

7. Final Probability Calculation:

   • Let P_HHT be the probability of HHT appearing first, and P_HTT be the probability of HTT appearing first.
   • Since these are mutually exclusive events, P_HHT + P_HTT = 1.
   • From recursive calculations, P_HHT = 2 * P_HTT.
   • Solving 2 * P_HTT + P_HTT = 1 gives P_HTT = 1/3 and P_HHT = 2/3.

Conclusion:

The sequence HHT is more likely to appear first than HTT when flipping a fair coin. The probability of HHT appearing first is 2/3, while the probability of HTT appearing first is 1/3. This results from the recursive nature of the sequences and how they reset based on the outcomes of subsequent flips.