2019 MI P2 Q10

A particle moves one step each time either to the right or downwards through a network of connected paths as shown above. The particle starts at \(S\), and, at each junction, randomly moves one step to the right with probability \(p\), or one step downwards with

probability \(q\), where \(q=1-p\) . The steps taken at each junction are independent. The particle finishes its journey at one of the \(6\) points labelled \({{A}_{i}}\) , where \(i=0,1,2,3,4,5\) (see diagram). Let \(\left\{ X=i \right\}\) be the event that the particle arrives at point \({{A}_{i}}\).

1. Show that\(\mathrm{P}\left( X=2 \right)=10{{p}^{2}}{{q}^{3}}\).[2]
2. After experimenting, it is found that the particle will end up at point \({{A}_{2}}\) most of the time. By considering the mode of \(X\) or otherwise, show that \(\frac{1}{3} < p < \frac{1}{2}\).[4]