First Success Distribution¶

Definition¶

Let $0<p\le1$ and $q=1-p$ $p$ . A discrete random variable $T$ has a first success distribution with parameter $p$ , written $T\sim\operatorname{FS}(p)$ $FS (p)$ , if $T$ counts the trial on which the first success occurs in independent trials with success probability $p$ :

\mathbb{P}(T=k)=q^{k-1}p,\qquad k=1,2,\ldots.

Interpretation¶

The first success distribution answers the question: in a sequence of independent trials with success probability $p$ , on which trial does the first success occur? The successful trial itself is counted, so $T$ takes the values $1,2,\ldots$ .

Probability bars for the trial of the first success with success probability 0.3, starting at 0.3 for the first trial and shrinking by a factor 0.7 at each step.

In the figure $p=0.3$ $0.3$ : the first bar is $\mathbb{P}(T=1)=p=0.3$ $0.3$ , and each later bar is $0.7$ times the previous one, $0.21$ , $0.147$ , and so on. The bars decrease from the start, so the most likely trial for the first success is always the first one, although the mean $1/p\approx3.33$ given in the figure lies further to the right.

The same shape as the geometric distribution, moved one step to the right because the successful trial is counted: $\mathbb{E}T=1/p$ $1/ p$ .

The formula reads as a story: $T=k$ happens if and only if the first $k-1$ trials fail and trial $k$ succeeds. Each trial is a Bernoulli variable, and by independence these $k$ results have probability $q^{k-1}p$ . In the same way $\mathbb{P}(T>k)=q^k$ $q^{k}$ , since $T>k$ means that the first $k$ trials all fail. The masses sum to $1$ by the geometric series $p(1+q+q^2+\cdots)=p/(1-q)=1$ $1$ , so a success eventually occurs with probability one.

Conventions differ between books and software. This page counts trials, while the geometric distribution $\operatorname{Geo}(p)$ counts the failures before the first success, so $T=G+1$ $1$ ; check which version a formula refers to, because the means differ by exactly one, $1/p$ against $q/p$ . Waiting for the $r$ -th success instead of the first leads to the negative binomial distribution, which is the first success distribution when $r=1$ $1$ .

Key Results¶

If $G\sim\operatorname{Geo}(p)$ $Geo (p)$ , then $T=G+1$ has the first success distribution.
$\mathbb{E}T=1/p$ and $\operatorname{Var}(T)=(1-p)/p^2$ $p) / p^{2}$ .

Examples¶

For $p=1/2$ $1/2$ , $\mathbb{P}(T=3)=1/8$ $1/8$ .
For $p=0.2$ $0.2$ , the expected trial number of the first success is $5$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3
Probability Theory for EOR · Week 4
Nicky van Foreest · Probability Distributions · Refresher

Aliases: first success random variable, one-based geometric distribution, FS(p), shifted geometric distribution Keywords: trial number of first success, waiting time to first success

Last updated: 20 October 2026 03:02