Maximize Your Guess with One Inquiry

Solution & Explanation

When tasked with predicting the outcome of 100 consecutive coin flips, the challenge is to maximize your chances of making an accurate prediction with just one yes-or-no question. Here's how you can approach the problem:

Understanding the Problem

• Total Possible Sequences: Each coin flip has 2 possible outcomes (heads or tails), and with 100 flips, there are $2^{100}$ possible sequences.
• Goal: Use a single question to halve the possible outcomes, thereby increasing the probability of correctly predicting the sequence.

Optimal Question Strategy

1. Question: "Is the first flip a head?"

   • Explanation: By asking this question, you immediately eliminate half of the total possible sequences. If the answer is "yes," you are left with sequences starting with heads, reducing the possibilities to $2^{99}$. If "no," you focus on sequences starting with tails, again reducing the possibilities to $2^{99}$.
   • Outcome: This simple question effectively reduces the complexity of the problem by half, maximizing the information gained from the question.

2. Alternative Question: "Do heads appear more than tails in the sequence?"

   • Explanation: This question divides the sequences based on the majority occurrence of heads or tails. If the answer is "yes," you know there are more heads than tails, narrowing down the possibilities to sequences that have a majority of heads.
   • Mathematical Consideration: This question does not reduce the possibilities to exactly $2^{99}$, but it provides valuable information about the distribution of heads and tails, which can be helpful in certain contexts.

Analyzing the Effectiveness

• Probability Reduction: Both questions aim to reduce the number of potential sequences from $2^{100}$ to $2^{99}$, effectively doubling your chances of guessing correctly.
• Information Gain: The question "Is the first flip a head?" provides immediate and straightforward information, making it a clear choice for maximizing information gain.
• Complexity: While the question regarding the distribution of heads and tails offers deeper insights, it might complicate the prediction process without necessarily providing a better probability of success.

Conclusion

In conclusion, the optimal strategy to maximize your chances of correctly predicting the sequence of 100 coin flips is to ask a question that effectively halves the number of possible sequences. The question "Is the first flip a head?" is simple, direct, and provides the maximum reduction in possibilities, ensuring the best chance of success with limited information.