Probability Tree & Contingency Table Calculator
Calculate multi-stage probability experiments, path probabilities (1st path rule), total probability (2nd path rule), Bayes conditional probabilities and synchronized contingency tables.
Baumdiagramm (2-Stufig)
Stochastisch abhängigVierfeldertafel (Wahrscheinlichkeiten)Summe Zeilen & Spalten = Randwahrscheinlichkeiten
| Ereignis | Krank (K) | Gesund (G) | Σ (Gesamt) |
|---|---|---|---|
| Test Positiv (+) | 0.09509.50% | 0.04504.50% | 0.1400P(Test Positiv (+)) |
| Test Negativ (-) | 0.00500.50% | 0.855085.50% | 0.8600P(Test Negativ (-)) |
| Σ (Gesamt) | 0.1000P(Krank (K)) | 0.9000P(Gesund (G)) | 1.0000 (100%) |
📚Schritt-für-Schritt Rechenweg
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Unfinished Games of Chance & Diagnostic Fallacies
In 1654, Chevalier de Méré asked Pascal: how should the prize money be divided fairly if a dice game is interrupted prematurely? Furthermore, how does one assess diagnostic test accuracy when a disease is exceptionally rare?
2. Tree Diagram Path Rules & Bayes' Theorem
Fermat and Pascal developed multiplication and addition path rules. Thomas Bayes formulated Bayes' theorem for conditional probability, demonstrating how prior beliefs (prior) are updated by new evidence (posterior).
3. Spam Filters, Medical Diagnostics & Forensic DNA
Naive Bayes email spam filters, AI diagnostic assistants in radiology, and legal forensic evidence evaluation rely on Bayesian conditional probability.
Real-World Applications of Probability Trees & Bayes
Explore how multi-stage probability trees empower data-driven decisions in medicine, engineering, and daily risk management.
Medical Screening & Bayes Paradox
Calculating positive predictive value under low population disease prevalence.
Even with 98% test accuracy, a rare 1% disease yields true positive probability of around 16%.
Manufacturing Defect Trees
Multi-stage production line failure rate analysis and defect origin tracking.
Determining total defect rate across multiple automated robotic assembly lines.
Supply Chain Risk Modeling
Combined stochastic delivery delays across multi-stop customs and weather hubs.
Predicting on-time delivery percentages for complex international logistics chains.
Guide: Probability Trees, Path Rules & Bayes Theorem
A probability tree diagram systematically structures sequential random events. It enables straightforward computation of intersection and marginal probabilities alongside contingency tables.
1st Path Rule (Multiplication Rule)
The probability of an entire branch path is the product of its individual probabilities: P(A ∩ B) = P(A) · P(B|A).
2nd Path Rule (Law of Total Probability)
To compute the overall probability of event B, add the probabilities of all mutually exclusive paths leading to B: P(B) = P(A ∩ B) + P(not A ∩ B).
Bayes Theorem
Calculates inverted conditional probability: P(A|B) = P(A ∩ B) / P(B) = (P(A) · P(B|A)) / P(B).
Stochastic Independence
Events A and B are stochastically independent if and only if P(A ∩ B) = P(A) · P(B).
Frequently Asked Questions about Probability Trees
What is the difference between replacement and without replacement?▼
With replacement, branch probabilities remain constant at every stage (independent draws). Without replacement, available item counts decrease, updating conditional probabilities dynamically.
When should I use a contingency table versus a tree diagram?▼
Contingency tables are optimal for static 2x2 categorical comparisons and fast marginal sum checks. Tree diagrams are best for chronological, sequential decision processes.
Why is Bayes Theorem crucial in medical diagnostics?▼
Because in low prevalence scenarios, false positives frequently outnumber true positives. Bayes Theorem computes the actual probability that a tested individual has the condition.
What is the intuitive meaning of stochastic independence?▼
Occurring of event A does not alter the likelihood of event B happening, meaning P(B|A) = P(B).
Wie hängen Baumdiagramm und Vierfeldertafel (2x2-Tabelle) zusammen?▼
Ein zweistufiges Baumdiagramm und eine Vierfeldertafel stellen dieselben stochastischen Zusammenhänge dar. Die Pfade des Baumdiagramms entsprechen den Schnittmengen P(A ∩ B) in den vier inneren Feldern der Vierfeldertafel. Aus der Vierfeldertafel lassen sich die Randwahrscheinlichkeiten P(A) und P(B) sowie die bedingten Wahrscheinlichkeiten P(B|A) = P(A ∩ B) / P(A) (Satz von Bayes) direkt ablesen und berechnen.