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  , 3, 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, 3, 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 - 7) f(m) - (-1 + p)  (2 m  - 17 m + 24) f(m - 3)

          2         2   2
     - 2 p  (-1 + p)  (m  - 9 m + 12) f(m - 4) + (m - 4) (m - 6) f(m - 6)

        2     2                          4
     - p  (2 m  - 21 m + 48) f(m - 7) + p  (m - 3) (m - 8) f(m - 8) = 0



                 subject to the appropriate initial conditions



                              and in Maple format



m*(-1+p)^4*(m-7)*f(m)-(-1+p)^2*(2*m^2-17*m+24)*f(m-3)-2*p^2*(-1+p)^2*(m^2-9*m+
12)*f(m-4)+(m-4)*(m-6)*f(m-6)-p^2*(2*m^2-21*m+48)*f(m-7)+p^4*(m-3)*(m-8)*f(m-8)
= 0


                 subject to the appropriate initial conditions