On the probability of the first player reaching m first if, starting at 0, a\

    t  each round they go one step to the right with probability , p

               or  , 1, 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, 1, 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                            2  4
m (-1 + p)  (m - 3) f(m) - (-1 + p)  (2 m  - 7 m + 4) f(m - 1) + (-2 m  p

          2  3        4      2  2         3      4        2      3    2      2
     + 4 m  p  + 8 m p  - 2 m  p  - 16 m p  - 4 p  + 8 m p  + 8 p  + m  - 4 p

                            2     2
     - 4 m + 4) f(m - 2) - p  (2 m  - 9 m + 8) f(m - 3)

        4
     + p  (m - 1) (m - 4) f(m - 4) = 0



                 subject to the appropriate initial conditions



                              and in Maple format



m*(-1+p)^4*(m-3)*f(m)-(-1+p)^2*(2*m^2-7*m+4)*f(m-1)+(-2*m^2*p^4+4*m^2*p^3+8*m*p
^4-2*m^2*p^2-16*m*p^3-4*p^4+8*m*p^2+8*p^3+m^2-4*p^2-4*m+4)*f(m-2)-p^2*(2*m^2-9*
m+8)*f(m-3)+p^4*(m-1)*(m-4)*f(m-4) = 0


                 subject to the appropriate initial conditions