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How does conditional probability change after new information about an event is known?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, conditional probability measures the likelihood of an event occurring given the knowledge that another event has already taken place. When new information (evidence) becomes available, it allows for the revision of initial predictions, a process central to Bayesian analysis.1234

This revision is mathematically formalized by Bayes’s theorem, which updates the "prior" probability of a hypothesis to a "posterior" probability based on the new evidence. The formula is expressed as:34

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

Where:
* $P (H)$ is the prior probability of the hypothesis.
* $P (E | H)$ is the probability of the evidence given the hypothesis is true.
* $P (E | \neg H)$ is the probability of the evidence given the hypothesis is false.

By incorporating new data, this method allows one to refine the probability of an event, moving from a general expectation to a more accurate assessment based on specific, observed outcomes.342