Jacob Bernoulli, Binomial Distribution, Probability, Central Limit Theorem, Deep Dives.
With each value of p is associated a probability density that can be read off from the inverse probability distribution function P(p|X_bar_n,n). If you know this inverse PDF, you can calculate the probability that p will lie within some specified interval [p_low, p_high], i.e. P(p_low ≤ p ≤ p_high)