Imagine I flip a coin 100 times in a sequence, and you need to guess the sequence. You are allowed to ask one yes/no question. What do you ask to maximize the probability of guessing the correct sequence
The Correct Answer and Explanation is:
To maximize the probability of guessing the correct sequence of 100 coin flips, the most effective yes/no question to ask would be:
“Is the sequence of coin flips symmetrical?”
Explanation:
The core idea behind this strategy is to leverage the structural properties of the sequence rather than attempting to guess each individual flip. Let’s break it down step by step:
- Symmetry of Sequences:
- A sequence is said to be symmetrical if it reads the same forward and backward (like a palindrome). For example, “HTTHT” is symmetrical, but “HTTHH” is not.
- The question directly probes the overall structure of the sequence. If the answer is “yes”, it means the sequence has a symmetry that can be exploited. If the answer is “no”, the sequence has no such symmetry, and you know you are dealing with a non-symmetric configuration.
- Probabilistic Advantage:
- If the sequence is symmetrical, the number of possible sequences reduces significantly. For instance, in a sequence of 100 flips, knowing that the sequence is symmetrical means you only need to guess half of the sequence, and the rest is automatically determined. For a 100-flip sequence, this halves the number of possibilities.
- If the answer is “no”, the sequence is non-symmetric, but by understanding this, you still eliminate half of the possible configurations, simplifying your task to guessing the exact arrangement from the remaining possibilities.
- Maximizing the Guessing Probability:
- By asking if the sequence is symmetrical, you cut the total number of possible outcomes in half. Since there are (2^{100}) possible sequences, the symmetry question effectively reduces the space of possible answers from (2^{100}) to (2^{99}), significantly increasing your chances of guessing the sequence correctly. Even though you still need to make the correct guess, the total set of possibilities has been narrowed down considerably.
In conclusion, asking whether the sequence is symmetrical is a strategic question that maximizes the probability of correctly guessing the sequence of coin flips, utilizing the structural property of symmetry to reduce the number of possibilities.