One of the reasons for opposing the death penalty is the probability of executing an innocent person. Suppose our policy choice regarding the death penalty has the objective of minimizing wrongful deaths, which is the sum of private murders and mistaken executions (which is tantamount to murder). A numerical example will help fix ideas: consider a population of 10 million identical persons. Over a given period, the probability of any one committing murder is = 0.00002; then the expected number of murders is 200. With a death penalty, the probability of comitting a murder drops, say to 0.00001; then the expected number of murders is 100. Suppose the probability of obtaining a conviction is 0.5, for which the probability of a mistaken execution is a high 0.1. Then without the death penalty, the expected number of wrongful deaths is 200 + 0 = 200; with the death penalty, the expected number of wrongful deaths is 100 + (100 * 0.5*0.1) = 105. The deterrent effect would have to be really small, i.e. a difference of 0.0000009 in the probability of committing murder, for the expected avoidable deaths to be equal between the with- and the without-death penalty scenarios. In short, wrongful execution requires an actual murder and a conviction, hence this will be a very small number relative to the population, and a small number even relative to the expected number of murders.
Put in another way: by not imposing the death penalty, the State is allowing 95 more wrongful deaths, but itself inflicting none. State without a death penalty would seem to be one in which the weight of a publicly-sanctioned wrongful death is much, much greater than the weight of a privately-committed wrongful death. Is there a good justification for such a weight?