Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 8 / b / i / Solution

For $X=m$, the first $m-1$ digits must all differ from $3$ and the next digit must be $3$. Independence givesThus $X$ has the geometric distribution counting trials up to and including the first success, with success probability $1/3$. For completeness, its mean follows from the tail-sum formula:The sum of the geometric series also proves finiteness of this expectation.