Coin tossing game with one yes/no question. Let's assume we are playing a game. You have $100$ coins, you toss them and do not tell me the results. I can ask one yes/no question and after your answer I'll start guessing the coin results. For each correct result I'll get $1$ dollar and for each wrong guess I'll lose $1$ dollar. What is the best strategy for me to play the game? What is the average income of the strategy? 
Edit: Here is a strategy with higher than $1$ dollar return. Ask whether number of heads are greater than or equal to tails. If the answer is yes guess all them to be heads otherwise guess all of them to be tails.
 A: Theoretically, the best way to get information is by splitting all possible sequences into two equal sized groups, so asking  "is the first flip heads?" yields the best chance. Asking if there are more heads than tails is not as helpful since it does not split the possibilities into two equal groups (one group contains the 50/50 split sequences).
This answer is still up for debate - see this link for some other possibilities.
https://www.glassdoor.com/Interview/Imagine-I-flip-100-coins-in-a-sequence-and-you-need-to-guess-the-sequence-You-are-allowed-to-ask-one-yes-no-question-Wha-QTN_1369290.htm, and you can find other takes and reasoning on this question there.
A: Another strategy is to ask if the first two flips are both heads.  If the answer is yes, which happens with probability $\frac 14$, you have an expectation of $+2$.  If the answer is no, which happens with probability $\frac 34$, you guess tails for both of the first two flips.  With probability $\frac 13$ after the answer you are $+2$ and with probability $\frac 23$ you are even.  In total you are $+1$ on average.  No better than asking about a single flip, but no worse, either.  This shows that dividing the possibilities evenly is not necessarily better than other choices.
