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

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.