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Ubify Financial Advisory Board
FINANCIAL ANALYSTS & CPA REVIEWERS
About This Tool
This statistical calculator computes revised posterior probabilities using Bayes' Theorem based on prior prevalence, true positive rates (sensitivity), and false positive rates. When evaluating clinical diagnostic screenings, fraud detection alerts, or spam classification algorithms, human intuition frequently succumbs to the base-rate fallacy; this tool determines the true mathematical likelihood of a condition given a positive test indicator.
### Bayesian Probability Formulas The calculator applies Bayes' rule in client state:
- **Prior Probability ($P(A)$):** Base-rate prevalence of the condition in the population. - **True Positive Sensitivity ($P(B|A)$):** Probability that the test indicates positive given the condition is present. - **False Positive Rate ($P(B|\neg A)$):** Probability that the test indicates positive when the condition is absent. - **Total Probability of a Positive Indicator ($P(B)$):** $$P(B) = [P(B|A) \times P(A)] + [P(B|\neg A) \times (1 - P(A))]$$ - **Posterior Probability ($P(A|B)$):** $$P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$$
### Worked Numeric Example Suppose a medical screening test for a rare disease with a 1.0% prevalence rate ($P(A) = 0.01$) has 95% sensitivity ($P(B|A) = 0.95$) and a 5% false positive rate ($P(B|\neg A) = 0.05$):
1. **True Positive Fraction:** $0.95 \times 0.01 = 0.0095$. 2. **False Positive Fraction:** $0.05 \times (1 - 0.01) = 0.05 \times 0.99 = 0.0495$. 3. **Total Positive Tests ($P(B)$):** $0.0095 + 0.0495 = 0.0590$ (5.9% of all tests). 4. **Posterior Probability ($P(A|B)$):** $0.0095 / 0.0590 \approx 0.1610$ (**16.1%**). 5. **Interpretation:** Despite a 95% accurate test, an individual receiving a positive result has only a **16.1% true probability** of actually having the condition because uninfected individuals vastly outnumber infected individuals in the broader population.
### Client-Side Processing Note All calculations run in client-side JavaScript. No medical records, screening statistics, or personal health inputs are transmitted across any network.
How to Use
- 1
Enter the prior probability P(A) representing condition prevalence in the population (%).
- 2
Specify the test sensitivity / true positive rate P(B|A) (%).
- 3
Enter the test false positive rate P(B|not A) (%).
- 4
Review the revised posterior probability P(A|B) and the total positive test rate breakdown.
Frequently Asked Questions
What is the base-rate fallacy in probability?
The base-rate fallacy occurs when people judge the likelihood of an outcome based solely on test accuracy while ignoring how rare the underlying condition is in the general population.
How does high test accuracy result in a low posterior probability?
When a condition is rare (e.g. 1%), even a small 5% false positive rate produces more false positives than true positives across a large screening cohort.
Are any health screening or statistical queries sent to remote servers?
No. All Bayesian equations evaluate locally in your web browser memory.