= A shared coin prevents a zero-one conclusion
Let $Z$ have the fair <Bernoulli distribution> and set $X_j=Z$ for every $j$. Each coordinate has the same fair <probability distribution> as in the independent model, but the coordinates are dependent: $\mathbb P(X_1=1,X_2=1)=1/2$, whereas the product of the marginal <probabilities> is $1/4$. Every future <sigma-algebra> equals $\sigma(Z)$, so the <tail sigma-algebra> also equals $\sigma(Z)$.
The <tail event> of infinitely many heads is now exactly $\{Z=1\}$, with <probability> $1/2$. For the finite-window <events> $B_{m,r}$ in <infinitely many heads as a tail event>,
$$
\begin{array}{c|cc}
&\text{independent fair coordinates}&\text{one shared fair coin}\\
\mathbb P(B_{m,r})&1-2^{-r}&1/2\\
\mathbb P(A)&1&1/2
\end{array}
$$
The <sample space> can even be the same full <Cartesian product> $\{0,1\}^{\mathbb N}$ in both models: the dependent <probability measure> assigns mass $1/2$ to each of the all-zero and all-one <sequences>. Then $A$ is the same literal <tail event> in both models. Only the <probability measure> and its <independence> properties change.
\Image[/tail-events-independent-and-shared-coin-probabilities.png]
{title=More independent trials increase the chance of a head; repeated copies of one coin do not}
{description=Exact constructed <probabilities> for at least one head in a window of $r$ coordinates. The independent curve is $1-2^{-r}$ and tends to one; the shared-coin curve stays at $1/2$. These are calculations from the two specified <probability measures>, not measurements or simulated frequencies.}
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