1.10 Bayes' Theorem¶

Overview:¶

Bayes' Theorem connects prior probabilities (initial beliefs) with posterior probabilities (updated beliefs) based on new evidence. It is a cornerstone of conditional probability.

\( P(A|B) \): Posterior probability of event A, given that event B has occurred.
\( P(A) \): Prior probability of event A.
\( P(B|A) \): Probability of event B occurring, given that event A has occurred.
\( P(B) \): Total probability of event B.

Application Example:¶

Scenario: Two movies—KGF (Event A) and Kantara (Event B).¶

\( P(A) \): Probability of KGF being a hit = 0.34.
\( P(B) \): Probability of Kantara being a hit = 0.60.
\( P(A ∩ B) \): Probability of both being hits = 0.18.

Calculate \( P(A|B) \) (KGF being a hit, given Kantara is a hit):¶

Interpretation: Knowing Kantara is a hit reduces the probability of KGF being a hit from 0.34 to 0.30.

Calculate \( P(B|A) \) (Kantara being a hit, given KGF is a hit):¶

Interpretation: Knowing KGF is a hit reduces the probability of Kantara being a hit from 0.60 to 0.53.

Insights from the Example:¶

Dependency: The events (KGF and Kantara being hits) are not independent, as the joint probability \( P(A ∩ B) \) does not equal \( P(A) x P(B) \).
Posterior vs. Prior: Posterior probabilities (\( P(A|B), P(B|A) \)) differ from prior probabilities (\( P(A), P(B) \)) due to event interdependence.

Generalization (Law of Total Probability):¶

Bayes' Theorem can be extended using the Law of Total Probability:

Mutually Exclusive and Collectively Exhaustive Events
Example: For Kantara revenue levels (4L, 5L, 6L):
- \( P(A) \): Probability of KGF being a hit.
- Break down \( P(A) \) into contributions from all revenue levels of Kantara.