Probability distribution for radiation from a black hole in the presence of incoming radiation

Using only results from quantum field theory in curved space-time, we derive the formula for the probability that $k$ particles in a given mode will emerge from a Kerr black hole if $j$ particles in that mode are sent in at late times. Our formula agrees with the formula previously derived by Page and by Bekenstein and Meisels using other arguments. In particular, this proves that a Schwarzschild black hole responds to incoming radiation exactly as a blackbody does.