Bayes' Theorem
Bayes' theorem, named after the 18th-century British mathematician Thomas Bayes, provides a way to update the probability of an event based on new evidence or information, describing conditional probability.
Basic Formulation
The basic form of Bayes' theorem is:
$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$
Where:
- $P(A|B)$ is the probability of A given B
- $P(B|A)$ is the probability of B given A
- $P(A)$ is the probability of A
- $P(B)$ is the probability of B
This formula allows us to calculate the probability of an event A occurring, given that we know event B has occurred, by using our prior knowledge of the probabilities of A and B, and the probability of B occurring given A.
Application to Multiple Events
Students are expected to apply Bayes' theorem to multiple events. This extension of the basic formula involves considering multiple conditional probabilities.
For $n$ events, Bayes' theorem is expressed as
