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Statistics • Probability, Bayes' Theorem & Odds

Probability, Bayes' Theorem & Odds

Calculate posterior probability with Bayes' Theorem, Birthday Paradox match chances, permutations nPr, combinations nCr, and lottery expected values.

Bayes' Theorem Posterior Probability Engine

Calculate true posterior probability P(A|B) given prior base rate, test sensitivity, and false positive rate.

Population Intuition (Per 10,000 People):
  • Condition Present: 100 people
  • True Positives: 99 people
  • False Positives: 495 people
Posterior Probability P(A|B)
16.67%
Total Positive Test Rate P(B):5.94%
False Discovery Rate:83.33%
P(A|B) = [P(B|A)·P(A)] / P(B)
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Calculation Guide & Reference

Bayesian Posterior Odds, Combinatorics & Lottery Mathematics

Calculate posterior probabilities using Bayes' Theorem, discover the Birthday Paradox matching probability, compute permutations P(n,r), and evaluate lottery expected values.

Standardized Mathematical Formula
P(A|B) = [ P(B|A) * P(A) ] / P(B)

Bayes' Theorem updates the prior degree of belief in event A given new empirical evidence B.

Variables:
P(A):Prior probability (base rate)
P(B|A):Likelihood / True positive sensitivity
P(B):Marginal probability of observed evidence
How It Works (Step-by-Step)
  • 1Select between Bayes' Theorem, Birthday Paradox, Permutations/Combinations, Coin Tosses, or Lottery EV.
  • 2Input parameters such as prevalence, test sensitivity, set sizes, or jackpot values.
  • 3Receive real-time posterior probabilities, combination counts, and expected financial returns.
Real-World Numerical Example
Disease Screening with 1% Prevalence and 99% Sensitive Test

Base rate is 1%, test has 99% sensitivity and 5% false positive rate.

•P(Disease) = 0.01; P(No Disease) = 0.99.
•P(Positive | Disease) = 0.99; P(Positive | No Disease) = 0.05.
•P(Positive) = (0.99 * 0.01) + (0.05 * 0.99) = 0.0099 + 0.0495 = 0.0594.
•P(Disease | Positive) = 0.0099 / 0.0594 = 16.67%.
Result: Even with a 99% accurate test, a positive result indicates only a 16.67% chance of having the condition due to low prior prevalence.
Calculation Best Practices & Tips
In medical screening and classification, prior prevalence dramatically affects posterior positive predictive values (Bayes’ Theorem).
Use Bessel’s correction (n - 1) when calculating sample standard deviation to avoid systematically underestimating population variance.
Check assumptions of normality and homoscedasticity before making inferences with linear regression and ANOVA.

Frequently Asked Questions (FAQ)

Because we compare pairs of people: C(23, 2) = 253 pairwise comparisons, each having a 364/365 chance of not sharing a birthday. (364/365)^253 ≈ 49.3% chance of no match, meaning a 50.7% chance of at least one match.

Permutations nPr count arrangements where order matters (e.g. race finishing positions); combinations nCr count selections where order does not matter (e.g. choosing a committee). nPr is always larger than nCr for the same n and r.

When a condition is rare, even a highly accurate test produces more false positives than true positives in absolute terms, because the large pool of healthy people generates more false alarms than the small pool of actually-sick people generates correct positives — exactly what Bayes' Theorem quantifies.

Expected Value = (Probability of Winning × Prize Amount) − Ticket Cost, summed across all possible prize tiers. A negative expected value means the game favors the house over the long run.

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