Conditional Probability Calculator

Conditional Probability is evaluated from P - Joint Probability, P - Probability of Event B and P - Probability of Event A. The calculation reports P - Probability of A given B, P as% and P - using Bayes.

Results

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About the Conditional Probability Calculator

Conditional Probability is treated here as a quantitative relation between P - Joint Probability, P - Probability of Event B, P - Probability of Event A and P - P and P - Probability of A given B, P as% and P - using Bayes.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
P(A|B) = P(A∩B) / P(B). For Bayes' theorem: requires P(A), P(B|A), and P(B).

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

P(A|B) = P(A∩B) / P(B). For Bayes' theorem: requires P(A), P(B|A), and P(B).

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Medical test: P(disease|positive test). Prevalence 1%, sensitivity 95%, false positive rate 5%

Inputs

p_a_and_b: 0.0095 p_b: 0.059 p_a: 0.01
P - Probability of A given B: 0.161. P as%: 16.1%. P - using Bayes: 0.95

With P - Joint Probability = 0.0095, P - Probability of Event B = 0.059 and P - Probability of Event A = 0.01 as the stated inputs, the result is P - Probability of A given B = 0.161, P as% = 16.1% and P - using Bayes = 0.95. Each value corresponds to the declared output fields.

Example 2: Email spam filter: P(spam|contains 'free money'). P(free money in spam)=0.8, P(free money)=0.10, P(spam)=0.20

Inputs

p_a_and_b: 0.16 p_b: 0.1 p_a: 0.2
P - Probability of A given B: 1.6. P as%: 160%. P - using Bayes: 0.8

With P - Joint Probability = 0.16, P - Probability of Event B = 0.1 and P - Probability of Event A = 0.2 as the stated inputs, the result is P - Probability of A given B = 1.6, P as% = 160% and P - using Bayes = 0.8. Each value corresponds to the declared output fields.

Example 3: Quality control: P(machine A made defective part). Machine A: 60% of production, 2% defective. Part is defective — which machine?

Inputs

p_a_and_b: 0.012 p_b: 0.018 p_a: 0.6
P - Probability of A given B: 0.6667. P as%: 66.67%. P - using Bayes: 0.02

With P - Joint Probability = 0.012, P - Probability of Event B = 0.018 and P - Probability of Event A = 0.6 as the stated inputs, the result is P - Probability of A given B = 0.6667, P as% = 66.67% and P - using Bayes = 0.02. Each value corresponds to the declared output fields.

Example 4: Weather forecasting: P(accident|rainy day). P(accident and rain) = 0.03, P(rain) = 0.15

Inputs

p_a_and_b: 0.03 p_b: 0.15 p_a: 0.05
P - Probability of A given B: 0.2. P as%: 20%. P - using Bayes: 0.6

With P - Joint Probability = 0.03, P - Probability of Event B = 0.15 and P - Probability of Event A = 0.05 as the stated inputs, the result is P - Probability of A given B = 0.2, P as% = 20% and P - using Bayes = 0.6. Each value corresponds to the declared output fields.

Common Use Cases

  • Calculate P(A|B) given joint and marginal probabilities
  • Apply Bayes' theorem to update probabilities
  • Medical diagnostic probability calculation