Kekkei - Advancing Financial Science For Everyone

Introduction

Two of the most powerful tools in probability: the Law of Total Probability (breaking problems into cases) and Bayes' Theorem (updating beliefs with evidence).

Learning Objectives:

Apply law of total probability
Use Bayes' theorem for inference
Solve real-world problems

Law of Total Probability

If $B_1, B_2, ..., B_n$ partition the sample space (disjoint and cover $\Omega$ ), then:

$P(A) = \sum_{i=1}^{n} P(A | B_i) P(B_i)$

Strategy: Break $A$ into cases based on which $B_i$ occurs.

Bayes' Theorem

$P(B_i | A) = \frac{P(A | B_i) P(B_i)}{P(A)} = \frac{P(A | B_i) P(B_i)}{\sum_j P(A | B_j) P(B_j)}$

Interpretation:

$P(B_i)$ : Prior probability (before seeing $A$ )
$P(B_i | A)$ : Posterior probability (after seeing $A$ )
$P(A | B_i)$ : Likelihood (how likely $A$ is under $B_i$ )

python

# Medical test example
def bayes_medical_test():
    # Prior: 1% have disease
    p_disease = 0.01
    p_no_disease = 0.99
    
    # Likelihood: test accuracy
    p_pos_given_disease = 0.95  # sensitivity
    p_pos_given_no_disease = 0.05  # false positive rate
    
    # Total probability of positive test
    p_pos = p_pos_given_disease * p_disease + p_pos_given_no_disease * p_no_disease
    
    # Bayes: P(disease | positive test)
    p_disease_given_pos = (p_pos_given_disease * p_disease) / p_pos
    
    print(f"P(disease) = {p_disease:.2%} (prior)")
    print(f"P(+ | disease) = {p_pos_given_disease:.2%}")
    print(f"P(+ | no disease) = {p_pos_given_no_disease:.2%}")
    print(f"\nP(disease | +) = {p_disease_given_pos:.2%} (posterior)")

bayes_medical_test()

Key Takeaways

Law of total probability: Sum over partition cases
Bayes' theorem: Update beliefs with evidence
Applications: Diagnostics, spam filtering, ML

Next: Independence of events!

IntroductionFocusStart Focus Mode

Law of Total ProbabilityFocusStart Focus Mode

Bayes' TheoremFocusStart Focus Mode

Key TakeawaysFocusStart Focus Mode

Introduction

Law of Total Probability

Bayes' Theorem

Key Takeaways