Bayes' Theorem of Probability With Tree Diagrams & Venn Diagrams

Introduction to Bayes' Theorem

  • Focus on Bayes' theorem and its applications.

Conditional Probability Formulas

  • Definition: The probability of event A occurring given that event B has occurred is denoted as P(A|B).

    • Formula: P(A|B) = P(A and B) / P(B)

  • Reverse Conditional Probability: The probability of event B given A is determined by:

    • Formula: P(B|A) = P(B and A) / P(A)

  • Equality of Joint Probabilities: P(A and B) = P(B and A)

  • Key relationship derived from these formulas:

    • P(A|B) * P(B) = P(B|A) * P(A)

Bayes' Theorem Derivation

  • Rearranging the joint probabilities leads to:

    • P(A|B) = [P(B|A) * P(A)] / P(B)

  • Usage of Bayes' Theorem:

    • Helps in calculating conditional probabilities in reverse scenarios.

Example: Probability Calculation Using Bayes' Theorem

  • Events Defined:

    • Event A: Drawing numbers 1, 2, 3, 4, and 5.

    • Event B: Drawing numbers 4, 5, 6, 7, 8, and 9.

  • Intersection of A and B:

    • Common numbers: 4 and 5.

  • Calculations Needed:

    • P(A) = 5/9 (numbers in A out of total)

    • P(B) = 6/9 (numbers in B out of total)

    • P(B|A) = 2/5 (numbers in A that are also in B)

  • Application of Bayes' Theorem:

    • Formula:

      • P(A|B) = [P(B|A) * P(A)] / P(B)

    • Calculation:

      • Substitute: P(A|B) = [2/5 * 5/9] / (6/9)

      • Simplifying results in P(A|B) = 1/3.

Confirmation Using Conditional Probability

  • Direct Calculation:

    • P(A and B) = 2/9 (numbers common to A and B)

    • P(B) = 6/9 (numbers in B)

    • Thus, P(A|B) = (2/9) / (6/9) = 2/6 = 1/3, confirming Bayes' theorem result.

Another Example: Medical Screening and Prostate Cancer

  • Problem Statement: Calculate the probability a man has cancer given a positive test result.

  • Given Information:

    • Probability of having cancer, P(C) = 12% = 0.12

    • Positive test result given cancer, P(Pos|C) = 95% = 0.95

    • Positive test result given no cancer, P(Pos|~C) = 6% = 0.06

  • Finding P(Pos):

    • Understanding rates and using a tree diagram:

      • P(C) = 0.12, P(~C) = 0.88 (1 - 0.12)

      • Calculate positive outcomes:

        • P(C and Pos) = P(C) * P(Pos|C) = 0.12 * 0.95 = 0.114

        • P(~C and Pos) = P(~C) * P(Pos|~C) = 0.88 * 0.06 = 0.0528

    • Total positive results: P(Pos) = 0.114 + 0.0528 = 0.1668

  • Final Calculation Using Bayes' Theorem:

    • Formula: P(C|Pos) = [P(Pos|C) * P(C)] / P(Pos)

    • Calculation: P(C|Pos) = [0.95 * 0.12] / 0.1668 = 0.68345.

    • Conclusion: Approximately 68.3% chance of having cancer given a positive test result.

Understanding False Positives

  • Evaluate the probability a man does not have cancer despite a positive test result:

    • 1 - P(C|Pos) = 100% - 68.3% = 31.7%.

Alternative Calculation Using Population Size

  • Population scenario analysis: Assuming 10,000 individuals:

    • With cancer: 1,200, without cancer: 8,800.

  • Positive outcomes among those with cancer: 1,140 (95% of 1,200).

  • Positive outcomes among those without cancer: 528 (6% of 8,800).

  • Total positive results: 1,668 (1,140 + 528).

    • Result reconfirms previous calculations: P(C|Pos) = 1,140 / 1,668 = 0.683.

Conclusion

  • Two methods to arrive at the same probability using Bayes' theorem and scenario modeling.

  • Key takeaway: Understanding conditional probabilities leads to better interpretation of real-world scenarios.