I want to describe a thought experiment to explore independent events, and whether they may actually be linked.
You are given a set of independent coin generators.
Coin generators are true random number generators that return 'Heads' or 'Tails'.
The probability that any of the independent coin generators would return heads is $0.5$. The set of all of these coin generators is $S$.
$\# S = n$.
You fire all the coin generators, and they all produce their results.
Before observing the coin generators, you split the coin generators into two groups $A$ and $B$.
The selection of each group is random. $\# A + \# B = n$
You get two pieces of paper and write out your expectation for the number of heads you expect for each group on each piece.
You then observe $A$.
Do you change your observation for $B$?
Should you?
The random variables of interest are ${\#}h(A)$ and ${\#}h(B)$.
A few things I feel need to be made explicit. As at the time you conducted the experiment (immediately after firing the coin generators and before splitting $S$), the number of heads in $S$ was fixed. Let's call the set of all heads in $S$ $H$.
For each $V \subset S$ the set of all heads in $V$ is $h(V)$.
The selection process is random, this means that $\frac{{\#}A}{{\#}S} : \frac{{\#}h(A)}{{\#}H}$ may be $\gt 1, \, \lt 1$ or $= 1$. However, for each $A$ and $S$, only a few selections ($n\choose {\#}A$) produce $\frac{{\#}A}{{\#}S} : \frac{{\#}h(A)}{{\#}H} = 1$.
$A = B'$ and vice versa.
Let's call the sampling from $S$ such that $\frac{{\#}A}{{\#}S} = \frac{{\#}h(A)}{{\#}H}$ the equal sampling $(T_E)$.
Let's call the sampling from $S$ such that $\frac{{\#}A}{{\#}S} \gt \frac{{\#}h(A)}{{\#}H}$ the $A$-biased sampling $(T_A)$.
Let's call the sampling from $S$ such that $\frac{{\#}A}{{\#}S} \lt \frac{{\#}h(A)}{{\#}H}$ the $B$-biased sampling $(T_B)$.
$Pr(T_E) = n{\choose}{\#}A$
$$Pr(T_A) = Pr(T_B) = \frac{2^n - {{n}\choose{\#}A}}{2}$$
I feel the above should be kept in mind when we consider whether $h(A)$ affects $E\left(h(B)\right)$. For one, observing a higher than expected amount of heads in $A$ may raise the probability of the $A$-biased sampling.
I think that depending on what I observed in $A$, that I may revise my beliefs about $B$. Assuming $n = 8$, ${\#}A = 6$, and $h(A) = 6$, I will update (an increase) my posterior probability on the selection chosen favouring $A$ over $B$ in distribution of heads. This may in turn lead me to lower my estimation of the heads contained in $B$.
I think my stance of possibly shifting my beliefs about $B$ based on my observation of $A$ is legitimate, as the selection process is random and may lead to an unfair distribution of heads.
Should we shift our beliefs after observing $A$ (and more importantly, why so)?
NOTE:
I think it is important to draw a distinction between this experiment and the gambler's fallacy. We do not first generate $A$ and then subsequently generate $B$ (in which case $h(A)$ would be independent of $h(B)$), we generate $S$. We do not observe the number of heads or tails in $S$. The coin generators may be fired in any order (temporal or otherwise), but we do not observe $S$ until after all $x \in S$ have finished generating.
We then randomly sample $S$ into $A$ and $B(A')$. After sampling $S$, we estimate the number of heads in $A$ and $B$ $\left(E\left(h(A)\right) \text{and} \, E\left(h(B)\right) \text{respectively}\right)$. After our estimation, we observe $A$: should we update $E\left(h(A)\right)$ in light of $h(A)$? I don't think it's as clear cut as people seem to be making it out to be.