Question

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

Answer 1

Is the number of heads in the first half of the sequence equal to the number of tails?

This question efficiently leverages the symmetry of a fair coin flip, maximizing information gain by narrowing down possibilities based on the balance of heads and tails in the sequence's halves.

This question is designed to extract maximum information with a single inquiry. If the answer is "yes," it implies that the number of heads and tails in the first half is the same. Consequently, the second half must also have an equal number of heads and tails for the overall sequence to be balanced. If the answer is "no," the opposite holds true.

This significantly narrows down the possibilities, reducing the potential sequences from 2^100 (which is astronomically large) to a much more manageable set. The strategy exploits the fact that a balanced coin sequence must have an equal number of heads and tails overall, and the balance is maintained in both halves.

This approach is efficient because it leverages the inherent symmetry of a fair coin flip. Without any prior knowledge of the specific outcomes, the question allows for a strategic division of possibilities, providing the best chance of guessing the correct sequence. In this way, the question optimizes the information gained, making it a powerful tool for narrowing down the possibilities with just one inquiry.

Complete question:

To maximize the probability of guessing the correct sequence, you could ask a question that helps you distinguish between two subsets of possibilities, ideally cutting the remaining possibilities in half. One such question could be:

"Does the number of heads in the first 50 flips equal the number of tails in the first 50 flips?"

Answer 2

There is no single yes/no question you can ask to significantly increase the probability of guessing a 100-flip coin sequence because each flip has an independent 50-50 chance of being heads or tails, and no question can change the randomness of the outcomes.

Step-by-step explanation:

To maximize the probability of guessing the correct sequence of a coin flipped 100 times, you want to ask a question that can eliminate half of the potential sequences. A good yes/no question could be: 'Is the first flip heads?'. However, given the nature of random sequences and the fact that each flip is independent with a 50-50 chance of being heads or tails, there isn't a question that can significantly increase your odds of guessing all 100 correctly because the number of potential sequences is 2100, which is a vastly huge number. Therefore, any single yes/no question will only provide information about one coin flip, which is negligible in the context of 100 flips. The only certain information you could potentially gain from one question pertains to one specific flip and does not affect the randomness of the other 99 outcomes.

The concept of probability and randomness indicates that while you can expect to get approximately 50 heads and 50 tails when you flip a fair coin many times, each individual flip is still independent and has an equal chance of being heads or tails. This illustrates that a sequence of flips is subject to random chance, and even with a biased coin as mentioned in the different scenarios, there is still a variability in the outcomes that manifests as randomness over many trials.