Bayes' Theorem: Intuition 1, Sets
What is Bayes' theorem and what is the intuition behind it?
Answer
The first intuition is thinking in terms of conditional probabilities (and sets):
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad \text{and} \quad
P(B \mid A) = \frac{P(A \cap B)}{P(A)}.
\]
Thus,
\[
P(A \cap B) = P(A \mid B)P(B) = P(B \mid A)P(A).
\]
Rearranging gives Bayes' theorem:
\[
P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}.
\]
Notes and comments
Comment 1: Note that in exercises for the Bayes' Theorem we use the LOTP for the denominator.