Under potential outcomes framework (Neyman-Rubin causal model), it is straighforward to show that difference-in-means is an unbiased estimator of average treatment effect under completely randomized trials (when we fix the number of experiment units in treatment/control groups).
However, in Bernoulli trials experiment, each unit is randomly assigned to treatment/control group with some fixed probability. The bias computation of difference-in-means estimator is more difficult because the number of units in control/treatment group is now a random variable.
Is the difference-in-means still unbiased for Bernoulli trials randomized experiments? Is the way to prove this using law of iterated expectation, by conditioning on the number of units in treatment group?