In designing clinical trials, power and sample size calculations are based on the effect size estimated from prior (often smaller) trials or sub-group analysis results. Such estimates are subject to randomness and variations. Consequently, the calculated sample size may not be adequate and it may be beneficial to re-estimate the effect size at some interim analyses and re-calculate the sample size. There are several methods for sample size modification in the literature, e.g., Proschan and Hunsburger (1995), Cui, Hung, Wang (1999), Muller, Schafer (2001), Chen, DeMets, Lan (2004), Gao, Ware, Mehta (2008) (and there are more). So, are these all valid methods? Which one is “better”? It turns out (Gao, Ware, Metha, 2008), all these methods can be explained using Markov process theory, and they are “equivalent” for different situations, with Gao, Ware, Mehta (2008) being the most general. Further, The theory of sequential testing, adaptive designs, phase 2/3 seamless combination, multiple dose comparison can all be unified using a simple tool from Markov process theory—transition density function, which is just the conditional probability density function (the Armitage- McPherson recursive algorithm can be derived using the transitional density function) . This unified approach includes sample and power calculation, exact p-values and confidence intervals for adaptive superiority and non-inferiority trials (including sub-population enrichment which does not require pre-specification of sub-populations); adaptive sequential designs, sample size and power calculation, conservative p-values and confidence intervals for phase 2/3 seamless combination; adaptive sequential designs, sample size and power calculation, exact p-values and conservative confidence intervals for multiple dose comparison (including sub-population enrichment with pre-specified sub-populations).