Day 7

Math 216: Statistical Thinking

Bastola

The Multiplicative Rule

  • The probability of the intersection of two events, \(A\) and \(B\), can be calculated using the multiplicative rule. This rule uses conditional probabilities:

    \[ P(A \cap B) = P(A) \cdot P(B|A) \]

    Alternatively:

    \[ P(A \cap B) = P(B) \cdot P(A|B) \]

  • Interpretation: The rule helps us understand how the occurrence of one event influences the probability of another.

Law of Total Probability

  • The law of total probability allows us to calculate the probability of an event by considering all possible scenarios. For two mutually exclusive and exhaustive events \(B\) and \(B^c\):

    \[ P(A) = P(B) \cdot P(A|B) + P(B^c) \cdot P(A|B^c) \]

  • Application: This is particularly useful when dealing with partitioned sample spaces.

Independence in Probability

  • Special case where knowledge of one event doesn’t affect another’s probability:

    \[ P(A|B) = P(A) \quad \text{and} \quad P(B|A) = P(B) \]

  • Simplified Intersection Rule:

    \[ P(A \cap B) = P(A) \cdot P(B) \]

  • Caution: Independence requires mathematical proof, not visual intuition from Venn diagrams

Bayes’s Rule: Inverting Probabilities

  • Derived from multiplicative rule symmetry:

    \[ P(A|B) = \frac{P(A) \cdot P(B|A)}{P(B)} \]

  • Enhanced with total probability law:

    \[ P(A|B) = \frac{P(A)P(B|A)}{P(A)P(B|A) + P(A^c)P(B|A^c)} \]

  • Enables belief updating with new evidence

Clinical Diagnostic Example

  • Practical Context: Assessing liver disease risk in alcoholic patients
  • Event Definitions:
    • \(A\): Liver disease (\(P(A) = 0.10\))
    • \(B\): Alcoholism (\(P(B) = 0.05\))
  • Clinical Data: \(P(B|A) = 0.07\) (Alcoholism rate among liver disease patients)

Bayesian Analysis in Practice

  • Objective: Calculate \(P(A | B)\) - disease probability given alcoholism

  • Bayesian Calculation:

    \[P(A | B) = \frac{P(A)P(B|A)}{P(B)} = \frac{0.10 \cdot 0.07}{0.05} = 0.14\]

  • Interpretation: Alcoholism doubles disease risk compared to baseline (10% → 14%)

Core Concepts Recap

  1. Multiplicative Rule: Joint probabilities through conditioning
  2. Independence: When information becomes irrelevant
  3. Total Probability: Divide-and-conquer strategy
  4. Bayesian Inference: Evidence-driven probability updating