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  , 5, 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, 5, 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 - 11) f(m) - (-1 + p)  (2 m  - 27 m + 60) f(m - 5)

          2         2   2
     - 2 p  (-1 + p)  (m  - 14 m + 30) f(m - 6) + (m - 6) (m - 10) f(m - 10)

        2     2                            4
     - p  (2 m  - 33 m + 120) f(m - 11) + p  (m - 5) (m - 12) f(m - 12) = 0



                 subject to the appropriate initial conditions



                              and in Maple format



m*(-1+p)^4*(m-11)*f(m)-(-1+p)^2*(2*m^2-27*m+60)*f(m-5)-2*p^2*(-1+p)^2*(m^2-14*m
+30)*f(m-6)+(m-6)*(m-10)*f(m-10)-p^2*(2*m^2-33*m+120)*f(m-11)+p^4*(m-5)*(m-12)*
f(m-12) = 0


                 subject to the appropriate initial conditions