# Practicalities of Reverse Engineering The Probability Distribution of a Bayesian Posterior

Starting with a discrete form of Bayes equation
$$p(H | E) = frac{p(E | H)p(H)} {p(H)p(E|H) + p(neg H)p(E|neg H)}$$

Where $$p(E|H) = 1$$, and $$p(neg H) = 1 – p(H)$$, so really we can simplify to

$$p(H | E) = frac{p(H)} {p(H) + p(E|neg H) – p(E|neg H)p(H)}$$

Goal: Choose two probability distributions, one for $$P(H)$$ and one for $$P(E|neg H)$$ such that $$P(H|E) < 0.5$$ in $$80%$$ of cases. $$P(E|neg H)$$ needs to be quite low, $$10%$$ or less.

It’s been a few years since I did stats, and I don’t think I ever did much beyond joint distributions where the random variables were simply multiplied together. Where does one start with a problem like this, and how practical is it?

Note: The purpose of this to create hypothetical decision making scenarios for students where they are likely to commit the base rate fallacy. $$E$$ stands for evidence and $$H$$ stands for hypothesis.