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Engineering Data Analysis2D

Conditional Probability - Theory & Concepts

Understanding how probabilities change when new information is available, including Bayes' Theorem and Independence.

Engineering Data Analysis

Bayes' Theorem & Diagnostic Testing

Visualize conditional probability and Bayes' theorem using a probability tree. Adjust the prior probability and test accuracy to see how they impact the posterior probability.

Prior Rate

P(D)

5.0%

Probability of a random component being defective.

True Positive

P(T|D)

90.0%

Sensitivity: Test is positive when defect is present.

False Positive

P(T|G)

10.0%

False Alarm: Test is positive when NO defect is present.

Probability Tree Diagram

P(D)

= 5.0%

P(G)

= 95.0%

P(T|D)

= 90.0%

P(\sim T|D)

= 10.0%

P(T|G)

= 10.0%

P(\sim T|G)

= 90.0%

Start

Defective (D)

Good (G)

Positive (T)

=4.50%

Negative (~T)

=0.50%

Positive (T)

=9.50%

Negative (~T)

=85.50%

Total Positives $P(T)$

14.00%

Sum of True Positives and False Positives

Posterior $P(D|T)$

32.1%

Prob. it is defective GIVEN a positive test

Bayes' Theorem Calculation

P(D|T) = \frac{P(T|D) \cdot P(D)}{P(T|D) \cdot P(D) + P(T|G) \cdot P(G)}

P(D|T) = \frac{0.90 \cdot 0.050}{0.90 \cdot 0.050 + 0.10 \cdot 0.950}

P(D|T) \approx 32.1\%

Introduction to Data Analysis - Theory & Concepts Introduction to Data Analysis - Theory & Concepts - Engineering Data Analysis Bias Precision Descriptive Statistics - Theory & Concepts Descriptive Statistics - Theory & Concepts - Engineering Data Analysis Box Whisker

Engineering Data Analysis

Probability Tree Diagram

Total Positives P(T)P(T)P(T)

Posterior P(D∣T)P(D|T)P(D∣T)

Bayes' Theorem Calculation

Related simulations

Total Positives $P(T)$

Posterior $P(D|T)$