N2022 P2 Q9

A game is played using a counter on a board.

• The counter starts from a point S at the bottom of the board and moves upwards (see diagram).

• When leaving S the counter is equally likely to move up to the left to junction L or up to the right to junction R.

• At every junction after S, the counter moves up to the left with probability \(p\) or up the right with probability \(q\), where \(p+q=1\).

• The counter eventually arrives at one of the endpoints A, B, C, D, E or F.

1. Show that the probability the counter arrives at B is \(2{{p}^{3}}q+\frac{1}{2}{{p}^{4}}\).[2]

John and Kath each play this game, and their counters both arrive at B.

2. Find, in terms of \(p\), a simplified fraction for the probability that Jon’s and Kath’s counters followed exactly the same route.[4]

The probability that a counter arrives at B is the same as the probability that a counter arrives at C.

3. Find the value of \(p\).[4]