# Bayes Theorem

Probability discovered by Thomas Bayes in the 18th century.

Probability of a proposition is the chance or likelihood that a proposition is true.

If 1 student in 20 has the flue, the probability is 1 in 20.

Prior:

$P(Sally\ has\ the\ flu) = 0.05$

Suppose there are 5 girls and 15 boys.

### Conditional Probabilities

Suppose we learn some more information about the situation.

The probability that Sally has the flu conditional on the patient being a girl is 0.2.

The probability that Sally has the flu conditional on the patient being a boy is 0.

We write this as:

$P(Sally\ has\ the\ flu\ |\ the\ flu\ patient\ is\ a\ girl) = 0.20$

$P(Sally\ has\ the\ flu\ |\ the\ flu\ patient\ is\ a\ boy) = 0$

**Sometimes you know that you might encounter some new evidence in the future, but you don't know how that evidence should effect the probability**

## Bayes Theorem

Bayes Theorem gives you a way to figure out what your conditional probabilities should be.

$P(H) =$ Prior probability, the probability that the hypothesis is correct.

$P(H|E) =$ Probability of hypothesis H, given condition E.

$P(E|H) =$ Probability of the condition E, given hypothesis E.

$P(E) =$ Probability of the condition E.

The probability of a hypothesis, $H$ conditional on a new piece of evidence, $E$

$P(H|E) = \frac{P(E|H)P(H)}{P(E)}$

To summarize, Bayes Theorem tells us how to calculate the probability of a hypothesis given a condition. This depends on three things:

- Conditional probability given H
- The prior probability of the hypothesis
- The prior probability of the evidence