Suppose there’s a sequence $a$ of 0 or 1, that is long enough , eg $length(a)=2^n$, $n$ sufficiently big enough integer. Now Alice is to guess the content of $a$.
If Alice knows nothing about $a$, in the worst case her guesses could be all wrong, or we note it as $\inf P=0$, where $P$ is the probability of Alice’s right guess.
Now let’s say Bob knows the content of $a$ and is allowed to help Alice with passing her an infinite series $b$ of 0 or 1, however, Alice could only see the first $k$ bits in $b$.
If $k=1$, Bob could help Alice to push $\inf P=50\%$, if they agreed beforehand that the first bit in $b$ , that is $b_1$ that could be either 0 or 1, indicates $a$ has at least half 0s or 1s.
Q1. How could Bob help Alice if $k=2$?
Q2. How could Bob help Alice generally for a given $k$?
Q3. Is there a strategy that as $k$ grows, Alice’s guesses could improve?
Of course there is a trifle strategy that set $b$ just as $a$, then if Alice could see the first $k$ bits, she could always make $k$ right guesses on $a$. But this strategy works quite bad as small $k$s, eg if $k=1$, Alice could only guarantee 1 bit to be guesses correctly , instead of $50\%$.
Is there a better strategy ?
Not a full solution, but too long for a comment.
Define the distance between two sequences $a$ and $b$ to be the number of places where they differ, or $d(a,b)=\sum|a_i-b_i|$. This is a metric on the space $A$ of binary sequences of length $N$. For $a\in A$ and $r\geq 0$, define the ball at $a$ and with radius $r$ by $B(a, r) = \{b\in A : d(b,a)\leq r\}$.
If $k$ is known by both Alice and Bob in advance, then there are $2^k$ different signals that Bob can send to Alice, and for each of them Alice will make a particular guess $a^{(j)}$, $j=1,\dots,2^k$. The question is then to place the $a^{(j)}$ such that $A$ is covered by $B(a^{(1)},r)\cup\dots\cup B(a^{(2^k)},r)$ with $r$ as small as possible.
For example, if $k=2$ and $N=4$, then Alice and Bob can agree that the four words $00,01,10,11$ should stand for the guesses $0000, 1001, 1110$, and $0111$, respectively. Then Alice is guaranteed a score of at least 75 %, as $A$ is covered by balls of radius 1 about the four given sequences, so she will make at most one mistake.
And if $k=1$ and $N$ is arbitrary, the two words $0$ and $1$ could stand for any sequence $a^{(1)}$ and its opposite sequence $a^{(2)}$.
But if $k$ is not known in advance, the problem seems less clear, as different strategies may work better for different values of $k$. Any strategy would then give Alice a nondecreasing sequence $p_1,p_2,\dots$, where $p_k$ is the guaranteed score that Alice can be sure of when $k$ is revealed, and there will surely not be any strategy which is best for all $k$.
If the strategies are ordered lexicographically, by looking at the first $p_k$ where they differ, then the problem becomes well defined, but it seems hard to find optimal strategies, except by brute force. It is possible, however, to solve the problem for each $k$ and to transmit the code words for each $k$ successively.