Lecture on Bayes' Theorem

Introduction

• Bayes' Theorem is not a new concept in probability.
• It involves recombining known information in a new way.
• Named after Mr. Bayes.

Example Scenario

• Event A: Having a disease.
• Event B: Testing positive for the disease.

Given Probabilities:

• 5% of people have the disease (P(A) = 0.05).
• If you have the disease, the probability of testing positive is high (0.99).
• 10% of people who don't have the disease test positive (false positives).

Bayes' Theorem Explanation

• Goal: Find the probability of having the disease given a positive test result (P(A|B)).
• Formula:
  • [ P(A|B) = \frac{P(A \cap B)}{P(B)} ]
• Multiplication Rule:
  • [ P(A \cap B) = P(A) \times P(B|A) ]
• Solution:
  • Calculate P(B) using the law of total probability:
    • [ P(B) = P(A \cap B) + P(\neg A \cap B) ]

Calculation Steps

1. Probability of A and B:
   • [ P(A \cap B) = P(A) \times P(B|A) = 0.05 \times 0.99 ]
2. Probability of Not A and B:
   • [ P(\neg A \cap B) = P(\neg A) \times P(B|\neg A) = 0.95 \times 0.10 ]
3. Combine to Find P(B):
   • [ P(B) = 0.0495 + 0.095 = 0.1445 ]
4. Find P(A|B):
   • [ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.0495}{0.1445} \approx 0.34256 ]

Insights

• About 14.45% of people test positive, yet only about 5% actually have the disease.
• Approximately one-third of positive tests are true positives.
• Most positive tests are false positives due to low disease prevalence.

Conclusion

• Bayes' Theorem can answer questions with seemingly insufficient information.
• Important for understanding probabilities in medical testing and other fields.
• No new probability rules; just a different arrangement of known concepts.

Closing

• Bayes' Theorem is widely used and valuable for probabilistic reasoning.
• End of lecture.