Best guess of 100 tosses after one yes-no query

A friend tosses a fair coin $100$ times and writes down the sequence. You may ask one question that the friend answers truthfully with yes or no, and then you must name the whole sequence. Your best achievable probability of being right has the form $a^{-b}$ with integers $a,b$ and $b$ as large as possible. Find $ab$.

$198$. Any yes/no question splits the $2^{100}$ sequences into a set $A$ and its complement; you then guess within the indicated part, succeeding with probability $\tfrac{|A|}{2^{100}}\cdot\tfrac1{|A|}+\tfrac{2^{100}-|A|}{2^{100}}\cdot\tfrac{1}{2^{100}-|A|}=2^{-99}$, whatever the question. So $a=2$, $b=99$.