= Memorylessness of the geometric distribution
{title2=$\mathbb P(T>m+r\mid T>m)=\mathbb P(T>r)$}
For the positive-integer <geometric distribution> with success <probability> $0<p\leq1$, the displayed identity holds for nonnegative <integers> $m,r$ whenever the conditioning <event> has positive <probability>. The first $m$ attempts have failed; future <independent> attempts use the same success <probability>. Equivalently, divide the tail $(1-p)^{m+r}$ by $(1-p)^m$. For a zero-based failures count $F=T-1$, use $\mathbb P(F\geq m+r\mid F\geq m)=\mathbb P(F\geq r)$. Constant success <probability> and <independence> are essential: accumulated wear or learning would require a different model.
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