Maybe I've misunderstood the question, but I'm wondering if JP is asking for something that cannot exist. Suppose a 'coin', simulated for example by a random-number generator on a computer, has a probability 'p' of coming up Heads. Then the probability of getting 'X' Heads out of 'n' is defined by a binomial formula involving 'p', 'X' and 'n'. However, if I observe 'X' heads occurring in 'n' tosses of the coin, I have a different piece of information - in fact less information. Even if X/n = p, I don't know that p is the probability of coming up Heads. So, in terms of knowledge, probability and statistics, I'm 'in a different place'. The case of getting HH on two coin tosses (with probability p^2) shows this. The case of getting H on one coin toss shows this too. If I could derive a binomial formula for the number of Heads in n' tosses of the coin, it would be equivalent to knowing what the probability of Heads is on each toss. But I don't know this, so I can't derive a binomial formula from observing a sequence of coin-tosses. Guy