Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S9 / 3 / a / Solution

After three multiplicative steps, $k$ upward moves and $3-k$ downward moves giveFor $k=0,1,2,3$, respectively, the prices are $\boxed{72.9,89.1,108.9,133.1}$. Since the steps are independent, the endpoint probabilities are binomial:In the displayed ascending-price order they are $(1-p{)}^3$, $3p(1-p{)}^2$, $3p^2(1-p)$, and $p^3$. Different paths with the same count of up moves recombine because multiplication by $1.1$ and $0.9$ commutes.