Pierre Simon Laplace, Inverse Probability, Central Limit Theorem, Probability, Deep Dives.
above formula and use your favorite library to calculate the probability density P(p | X_bar_n = x, n) for different values of p in the interval [0, 1], you’ll get the following plot. I used Python and scipy, specifically scipy.stats.binom and scipy.special.betainc. The PDF for P(p|X_bar_n=251527)