For the Bayes' prior probability (Beta(1,1)), the posterior probability is:  For the Jeffreys' prior probability (Beta(1/2,1/2)), the posterior probability is:  and for the Haldane prior probability (Beta(0,0)), the posterior probability is:  From the above expressions it follows that for "s"/"n" = 1/2) all the above three prior probabilities result in the identical location for the posterior probability mean = mode = 1/2. For "s"/"n"  mean for Jeffreys prior > mean for Haldane prior. For "s"/"n" > 1/2 the order of these inequalities is reversed such that the Haldane prior probability results in the largest posterior mean. The "Haldane" prior probability Beta(0,0) results in a posterior probability density with "mean" (the expected value for the probability of success in the "next" trial) identical to the ratio "s"/"n" of the number of successes to the total number of trials. Therefore, the Haldane prior results in a posterior probability with expected value in the next trial equal to the maximum likelihood. The "Bayes" prior probability Beta(1,1) results in a posterior probability density with "mode" identical to the ratio "s"/"n" (the maximum likelihood). In the case that 100% of the trials have been successful "s" = "n", the "Bayes" prior probability Beta(1,1) results in a posterior expected value equal to the rule of succession ("n" + 1)/("n" + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of 1 (absolute certainty of success in the next trial). Jeffreys prior probability results in a posterior expected value equal to ("n" + 1/2)/("n" + 1). Perks (p. 303) points out: "This provides a new rule of succession and expresses a 'reasonable' position to take up, namely, that after an unbroken run of n successes we assume a probability for the next trial equivalent to the assumption that we are about half-way through an average run, i.e. that we expect a failure once in (2"n" + 2) trials. The Bayes–Laplace rule implies that we are about at the end of an average run or that we expect a failure once in ("n" + 2) trials.