Geometric Waiting Time

An experiment is repeated independently and identically (i.i.d.). At each trial, event $A$ occurs with probability $p = \mathbb{P}(A)$ , with $0 < p \le 1$ . Define the waiting time

T = \inf\{n \ge 1:\ \text{$A$ occurs at the $n$-th trial}\}.

General question: Show that $T$ follows a geometric distribution and compute $\mathbb{P}(T = n)$ for $n \ge 1$ and $\mathbb{E}[T]$ .
Application (die roll): How many rolls are needed on average to get a 4 for the first time?

Geometric Waiting Time

Answer