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  , 2, 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, 2, 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 - 5) f(m) - 2 (-1 + p)  (m  - 6 m + 6) f(m - 2)

        2         2     2
     - p  (-1 + p)  (2 m  - 13 m + 12) f(m - 3) + (m - 3) (m - 4) f(m - 4)

        2     2                          4
     - p  (2 m  - 15 m + 24) f(m - 5) + p  (m - 2) (m - 6) f(m - 6) = 0



                 subject to the appropriate initial conditions



                              and in Maple format



m*(-1+p)^4*(m-5)*f(m)-2*(-1+p)^2*(m^2-6*m+6)*f(m-2)-p^2*(-1+p)^2*(2*m^2-13*m+12
)*f(m-3)+(m-3)*(m-4)*f(m-4)-p^2*(2*m^2-15*m+24)*f(m-5)+p^4*(m-2)*(m-6)*f(m-6) =
0


                 subject to the appropriate initial conditions