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.
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 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: