= Truncated geometric distribution
{title2=$T\mid T\leq m$}
A truncated geometric distribution is obtained by conditioning a <geometric random variable> to lie in a specified finite range. For the trials-until-success convention $\Pr(T=k)=p(1-p)^{k-1}$ and an integer $m\geq1$,
$$
\Pr(T=k\mid T\leq m)=\frac{p(1-p)^{k-1}}{1-(1-p)^m},\qquad 1\leq k\leq m.
$$
The denominator is the <probability> of at least one success in the first $m$ independent <Bernoulli trials>. It renormalizes the retained probabilities.
For $p=1/2$ and $m=3$, the conditional probabilities are $4/7,2/7,1/7$. This differs from censoring or capping at $m$: the variable $\min(T,3)$ has probabilities $1/2,1/4,1/4$, because all later successes accumulate at three. Specifying conditioning rather than capping is essential in models of bounded retries. As $m\to\infty$, the conditioned probabilities tend to the original <geometric distribution>.
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