Say we have a random variable A that expresses the probability of heads in a coin. The PDF of A is 5a^4 if a between 0 and 1, and zero otherwise. The tosses are independent. Now say, I'm looking at two different events: B and C. Say B is the event that the first toss is heads. Say C is the event that the second toss is heads. If we want, we could introduce a little helper called D, which is the random variable that expresses the Bernoulli of a coin toss. I calculated that the probability of the event B is 5/6, because that's the result of the integration of the PMF of D in the possible range of values for A, between 0 and 1. This is correct. I also think that if I want to know the probability of event C given event B occured, I could just argue with independence of the tosses and so I would get the same result as for the pure probability of event B - is this assumption correct