Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 30 / 5 / b / v / Solution

Take the event to mean three smoking-free weeks in succession, $Y_{i1}=Y_{i2}=Y_{i3}=1$. The initial cumulative count is zero. Along this path it equals $0,1,2$ in weeks one, two, three, respectively. In the cumulative-response logistic model, defineThe chain rule for probabilities then givesThe three conditional probabilities are approximately $0.19891,0.27015,0.35559$. If “stop during the first three weeks” instead means at least one smoking-free week by week three, its different event has probability $1-(1-{\pi }_0{)}^3\simeq 0.48590$: along the all-failure path the cumulative count stays zero. Stating the event resolves this wording ambiguity.