Randomized designs are often used in clinical trials. In the literature, the power and sample size are usually obtained by ignoring the randomness of the allocation in randomized designs. However, when using a randomized design, the power is a random variable for a fixed sample size n. In this talk, we focus on the power function (random) and the sample size of two-arm (drug versus control) randomized clinical trials. We first give an example where a target power can not be achieved with high probability when the requisite sample size (based on the formula in the literature) is used. Then we obtain the power function for any given sample size and study the properties of this power function. Based on the power function, a formula of sample size is derived for randomized designs. This formula is applied to several randomization procedures. We also discuss our finding that response adaptive designs can be used to reduce the requisite sample size. Some simulation studies are reported.