A game is played using a counter on a board. • The counter starts from a point S at the bottom of th

Question

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. 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. 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. Find the value of $p$. [4]

Accepted Answer

(a) Shown (b) $\frac{5p^2 - 8p + 4}{(4-3p)^2}$ (c) $p = 0.710$

N2022 P2 Q9

N2022 P2 Q9

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