The Ok Corral and the Power of the Law (A Curious Poisson-Kernel Formula for a Parabolic Equation)

Abstract

Two lines of gunmen face each other, there being initially m on one side, n on the other. Each person involved is a hopeless shot, but keeps firing at the enemy until either he himself is killed or there is no one left on the other side. Let μ(m, n) be the expected number of survivors. Clearly, we have boundary conditions: μ ( m , 0 ) = m , μ ( 0 , n ) = n . We also have the equation μ ( m , n ) = m m + n μ ( m , n − 1 ) + n m + n μ ( m − 1 , n ) ( m , n ⩾ 1 ) . This is because the probability that the first successful shot is made by the side with m gunmen is m/(m + n). On using the recurrence relation (1.2) together with the boundary condition (1.1), the computer produces Table 1 below, in which m = 8192 + k , n = 8192 − k , d ( m , n ) = ( m 2 − n 2 ) = 128 ( 2 k ) . 1991 Mathematics Subject Classification 60F05.