Conditional Probability Calculator - Bayes Theorem Calculator & Posterior Probability Calculator
Free conditional probability calculator & Bayes Theorem calculator. Calculate P(A|B) with prior probability, likelihood, and posterior probability. Our calculator uses Bayes' Theorem to update probabilities based on new evidence and calculate conditional probabilities for statistical analysis.
Last updated: December 15, 2024
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Probability of event A (between 0 and 1)
Probability of event B (between 0 and 1)
Probability of B given A (between 0 and 1)
Conditional Probability Result
P(A|B) - Posterior Probability
0.2857
Formula:
P(A|B) = P(B|A) × P(A) / P(B)
Calculation Steps:
- Given: P(A) = 0.2500, P(B) = 0.3500, P(B|A) = 0.4000
- Apply Bayes' Theorem: P(A|B) = P(B|A) × P(A) / P(B)
- Substitute: P(A|B) = 0.4000 × 0.2500 / 0.3500
- Calculate: P(A|B) = 0.1000 / 0.3500 = 0.2857
Interpretation:
Conditional probability is higher than prior probability - event A is more likely given B
Bayes' Theorem:
- • Updates probability based on new evidence
- • P(A|B) is the posterior probability
- • P(A) is the prior probability
- • P(B|A) is the likelihood
Conditional Probability Calculator Types & Applications
Formula
P(A|B) = P(B|A) × P(A) / P(B)
Updates probability based on new evidence
Application
Bayesian Inference
Updates beliefs based on new data
Concept
Initial Belief
Starting probability before new evidence
Measure
P(B|A)
Probability of evidence given hypothesis
Application
Diagnostic Testing
Calculate true positive and false positive rates
Analysis
Probability Analysis
Multiple statistical probability measures
Quick Example Result
Given: P(A) = 0.25, P(B) = 0.35, P(B|A) = 0.40
P(A|B) =
0.2857
(28.57% probability)
How Our Conditional Probability Calculator Works
Our conditional probability calculator uses Bayes' Theorem to calculate the probability of an event A occurring given that event B has occurred. The calculation combines prior probability, likelihood, and evidence using the formula: P(A|B) = P(B|A) × P(A) / P(B).
Bayes' Theorem Formula
Formula:
P(A|B) = P(B|A) × P(A) / P(B)Where:
- P(A|B) = Posterior probability (probability of A given B)
- P(B|A) = Likelihood (probability of B given A)
- P(A) = Prior probability (initial probability of A)
- P(B) = Evidence (marginal probability of B)
This fundamental theorem updates probabilities based on new evidence, making it essential for statistical inference, medical diagnosis, and decision-making under uncertainty.
Mathematical Foundation
Conditional probability is a fundamental concept in probability theory. It quantifies how the probability of an event changes when we have information about another event. Bayes' Theorem provides a mathematically rigorous way to update probabilities based on evidence, making it the cornerstone of Bayesian inference and statistical reasoning.
- Conditional probability P(A|B) measures probability of A given B has occurred
- Bayes' Theorem relates conditional probabilities in both directions
- Prior probability is your initial belief before seeing evidence
- Posterior probability is the updated belief after seeing evidence
- Likelihood measures how probable the evidence is under the hypothesis
- Independent events have P(A|B) = P(A), meaning B doesn't affect A
Sources & References
- Introduction to Probability Theory and Its Applications - William FellerClassic reference for probability theory and Bayes' Theorem
- Pattern Recognition and Machine Learning - Christopher M. BishopComprehensive coverage of Bayesian methods and conditional probability
- Khan Academy - Probability and StatisticsFree educational resources for conditional probability and Bayes' Theorem
Need help with other statistical calculations? Check out our Bayes' theorem calculator and variance calculator.
Get Custom Calculator for Your PlatformConditional Probability Calculator Examples
Medical Test Information:
- Disease prevalence P(D): 0.01 (1%)
- Test sensitivity P(T|D): 0.95 (95%)
- Test specificity P(¬T|¬D): 0.99 (99%)
- Goal: Find P(D|T)
Calculation Steps:
- Calculate P(T|¬D) = 1 - 0.99 = 0.01
- Calculate P(T) = 0.95 × 0.01 + 0.01 × 0.99 = 0.0194
- Apply Bayes' Theorem: P(D|T) = 0.95 × 0.01 / 0.0194
- Result: P(D|T) ≈ 0.489 (48.9%)
Probability of Disease Given Positive Test: 48.9%
Despite a 95% sensitive test, the low prevalence (1%) means only 48.9% of positive tests are true positives. This demonstrates the importance of conditional probability in medical diagnosis.
Simple Example
P(A) = 0.5, P(B) = 0.7, P(B|A) = 0.6
P(A|B) = 0.6 × 0.5 / 0.7 = 0.4286
Weather Example
P(Clouds) = 0.4, P(Rain) = 0.2, P(Clouds|Rain) = 0.9
P(Rain|Clouds) = 0.9 × 0.2 / 0.4 = 0.45
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