Truncated geometric distribution ($T\mid T\le m$)

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 $\mathrm{Pr}(T=k)=p(1-p{)}^{k-1}$ and an integer $m\ge 1$,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.