Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / S / S2 / a / ii / Solution

For one trial and any measurable set $A$,Since the acceptance probability is $1/M$, conditioning gives target probability ${\int }_{A}p$. Repeating failed independent trials does not alter this conditional law: summing over the trial number multiplies by the geometric sum ${\sum }_{j\ge 0}(1-1/M{)}^j/M=1$. Taking $A=(-\infty ,x)$ proves