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*.