Bayes' formula: updating a probability with new information

In short

• The prior $P(E)$ is your belief before the news; the posterior $P(E \mid I)$ is your belief after it.
• The likelihood $P(I \mid E)$ is the probability of seeing the information if the event is true.
• The unconditional probability of the information $P(I)$ comes from the total probability rule over all events.
• Posterior = [likelihood ÷ $P(I)$] × prior, which is the same as the event's joint probability divided by the total.
• If the likelihood is above $P(I)$, the posterior rises above the prior; if it is below, the posterior falls.