= Infinitely many heads as a tail event
{title2=$A=\{X_j=1\text{ infinitely often}\}$}
On $\{0,1\}^{\mathbb N}$, the <event> of infinitely many heads is
$$
A=\bigcap_{m=1}^{\infty}\bigcup_{j\geq m}\{X_j=1\}.
$$
It says that after every index there is another head. For each $n$, the same <intersection> can start at $m=n$, so $A\in\sigma(X_n,X_{n+1},\ldots)$; therefore $A$ is a <tail event>. This verifies <measurability> directly using countable <unions> and <intersections>.
\Image[/tail-events-finite-prefix-and-infinite-heads.png]
{title=A finite prefix admits both infinitely many heads and only finitely many heads}
{description=The two illustrated extensions have the same initial observations. An alternating continuation belongs to the <tail event> of infinitely many heads; an all-tails continuation does not. Changing the highlighted finite prefix cannot change that infinite property, but it can change the exact first-head time.}
Assume now that the coordinates are fair <independent> <Bernoulli trials>. For fixed $m$, let $B_{m,r}$ be the <event> of at least one head among $X_m,\ldots,X_{m+r-1}$. Its <complement> requires $r$ tails, so
$$
\mathbb P(B_{m,r})=1-2^{-r},\qquad
B_m=\bigcup_{r\geq1}B_{m,r},\qquad
\mathbb P(B_m)=1.
$$
The last equality follows from <continuity of probability>. Now $A=\bigcap_{m\geq1}B_m$, a countable <intersection> of <almost sure events>, and hence $\mathbb P(A)=1$. This determines which of the two possibilities in <Kolmogorov's zero–one law> occurs. Equivalently, the <Borel-Cantelli second lemma> applies to the <independent events> $\{X_j=1\}$.
Nevertheless, no finite prefix certifies membership in $A$: every finite record can be extended by all tails or by alternating heads and tails. Under the fair <independent> model, conditioning on any exact finite prefix leaves $\mathbb P(A)=1$, since $A$ is independent of the initial <sigma-algebra>. The excluded all-tails continuation has <probability> zero, but remains a possible member of the <sample space>. Logical determination by a finite record and a conclusion holding <almost surely> are different statements.
Back to article page
