Lesson 3

A natural generalization of the question we considered above: suppose some experiment ends in success with probability \(p\). What is the expected number of repetitions until the first success?

Let us denote the corresponding variable (the number of repetitions until the first success) as \(\alpha\). Its distribution is called geometric: \(\alpha \sim \operatorname{Geo}(p)\). The name comes from the fact that its probability mass function forms a geometric progression: \[\Pr[\alpha=k]=(1-p)^{k-1}p \ .\] This means: for success to occur for the first time on step \(k\), there must first be \(k-1\) steps ending in failure, followed by one successful step. It is easy to verify (using the formula for the sum of a geometric series) that \(\sum_{k=1}^{\infty}\Pr[\alpha=k]=1\).