On the probability of the first player reaching m first if, starting at 0, at  each round they go one step to the right with probability , p

               or  , 4, steps to the left with probabilty, 1 - p



                              By Shalosh B. Ekhad



Consider  a game where players take turns and at each step either move  righ\

    t one unit, with probability, p

                or move, 4, units left with probability , 1 - p



        and whoever reaches the location m first is declared the winner.

What can you say about the probability of the first player of winning the ga\
    me?



           Theorem: The probabability of the first player winning is



                                 1/2 + 1/2 f(m)



                     where f(m) satisfies the  recurrence



          4                          2   2
m (-1 + p)  (m - 9) f(m) - 2 (-1 + p)  (m  - 11 m + 20) f(m - 4)

        2         2     2
     - p  (-1 + p)  (2 m  - 23 m + 40) f(m - 5) + (m - 5) (m - 8) f(m - 8)

        2     2                          4
     - p  (2 m  - 27 m + 80) f(m - 9) + p  (m - 4) (m - 10) f(m - 10) = 0



                 subject to the appropriate initial conditions



                              and in Maple format



m*(-1+p)^4*(m-9)*f(m)-2*(-1+p)^2*(m^2-11*m+20)*f(m-4)-p^2*(-1+p)^2*(2*m^2-23*m+
40)*f(m-5)+(m-5)*(m-8)*f(m-8)-p^2*(2*m^2-27*m+80)*f(m-9)+p^4*(m-4)*(m-10)*f(m-\
10) = 0


                 subject to the appropriate initial conditions