Probability Calculator
Analyze operational risk and business scenarios. Calculate single event probabilities, combined probabilities, complements, and use Bayes' Theorem for conditional probabilities. Assess defect rates, delivery risk, and process failure scenarios.
Analyze probabilities for risk assessment and scenario planning. Calculate simple events, combined probabilities, complements, or use Bayes' Theorem for conditional probabilities.
Number of successful/favorable outcomes
Total possible outcomes in the sample space
Formula
P(A) = Favorable Outcomes / Total Outcomes | P(A and B) = P(A) × P(B) [independent] | P(A or B) = P(A) + P(B) − P(A) × P(B) | P(not A) = 1 − P(A) | P(A|B) = [P(B|A) × P(A)] / P(B)
Probability is the likelihood an event occurs, from 0 (impossible) to 1 (certain). For independent events, multiply probabilities. For 'or', use inclusion-exclusion. The complement is 1 minus the probability. Bayes' Theorem updates probabilities based on new evidence.
Worked Example: Manufacturing Defect Risk
A factory produces 1,000 units daily. History shows 30 units are defective. What is the defect rate?
P(defect) = 30 / 1,000 = 0.03 = 3%
P(not defective) = 1 − 0.03 = 0.97 = 97%
A 3% defect rate means roughly 97% of units meet quality. For delivery risk: if two steps each have 95% on-time rates and are independent, P(both on time) = 0.95 × 0.95 = 0.9025 = 90.25% on-time delivery end-to-end.
Common mistakes to avoid
• Confusing independent with dependent events
Check if one event affects the other. Defect in part A and defect in part B may be independent if produced separately. But if part B fails due to part A failing, they are dependent.
• Multiplying probabilities for "or" instead of adding
For "A or B", use P(A) + P(B) − P(A)×P(B) for independent events. Multiplying gives the wrong result.
• Using raw outcomes instead of probabilities
Convert to decimal probabilities (0 to 1) before using combined formulas. If 3 out of 10, use 0.3, not 3/10 as separate values.
• Ignoring the base rate in Bayes’ Theorem
P(A|B) depends on P(A) (the prior). A test that is 99% accurate can give misleading results if the base rate of the condition is very low.
Frequently Asked Questions
Use this in your workflow
Use the Sample Size Calculator to determine required sample sizes for statistical validity of probability estimates. Use the Confidence Interval Calculator to quantify uncertainty in your probability estimates. Use the Z-Score Calculator for normal distribution tail probabilities. Use the Percent Change Calculator to track how probabilities change over time. Browse all Free Business Calculators.
When to use this calculator
- →Assessing defect rates, quality risks, or failure probabilities in manufacturing or operations
- →Calculating end-to-end delivery risk across multiple process steps or suppliers
- →Modeling scenario probabilities for strategic planning, capital investment, or contingency prioritization
- →Evaluating test or audit results using Bayes' Theorem to adjust confidence in decisions
- →Analyzing "at least one" problems — e.g., probability of any failure across a large set of tasks
Worked example 1: Defect rate and quality risk
A useful reference before entering your own figures above.
| Scenario | Calculation | Result |
|---|---|---|
| Defect rate | 30 defects / 1,000 units | 3% |
| Quality (no defect) | 1 − 0.03 | 97% |
| Two units both defective (independent) | 0.03 × 0.03 | 0.09% |
| At least one defect in 100 units | 1 − 0.97^100 | 95% |
A 3% defect rate seems low, but across 100 units, there's a 95% chance of encountering at least one defect. This illustrates why even low individual failure rates compound into significant risk across large populations.
Worked example 2: End-to-end delivery risk
Calculating risk across multiple supply chain steps.
| Step | On-Time Rate | Delay Risk |
|---|---|---|
| Supplier ships | 95% | 5% |
| Transport completes | 95% | 5% |
| Warehouse receives | 95% | 5% |
| End-to-end (independent) | 0.95³ = 86% | 14% |
Even with 95% on-time rates at each step, the combined probability of all three steps being on time is only 86%. The end-to-end delay risk is 14%. This shows the importance of process redundancy, buffers, or fallback suppliers for critical shipments.
Limitations and considerations
Probabilities are only as accurate as your historical data or assumptions. Past defect rates may not reflect future conditions due to process changes, new suppliers, or market shifts. Bayes' Theorem results are sensitive to the accuracy of all inputs — small errors in base rates or test accuracy can lead to misguided decisions. These calculations assume independence unless otherwise specified — if events are correlated (e.g., two defects from the same cause), results will be inaccurate. These results are for planning and illustrative purposes only — not recommendations.
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Frequently asked questions
What is probability?
Probability is a measure of likelihood, from 0 (impossible) to 1 (certain). In decimal form, 0.5 = 50% chance. In fractions, 1/2 = 50%. Probabilities quantify uncertainty and are essential for risk assessment, scenario planning, and decision-making.
How do I calculate probability from historical data?
Divide the number of times an event occurred by the total number of observations. Example: if you shipped 1,000 orders and 50 arrived late, the late-delivery probability is 50/1,000 = 0.05 = 5%. Use this as your baseline for future risk estimates.
What does "independent events" mean?
Events are independent if one does not affect the other. Two coin flips are independent — the result of the first doesn't change the second. Two production defects are often independent unless caused by the same root issue. For independent events, multiply probabilities. For dependent events, use conditional probabilities.
How do I use Bayes' Theorem in practice?
Bayes' Theorem updates the probability of something based on new evidence. Example: your quality test is 99% accurate. A unit tests positive for defects. What's the probability it's actually defective? If only 1% of units are defective (prior), Bayes' Theorem shows the actual probability might be only ~50%, not 99%. Always consider the base rate.
How do I reduce calculated risk?
Reduce individual failure rates through better process control, training, or equipment. Introduce redundancy — backup suppliers, parallel steps, buffer stock. Decouple processes so one failure doesn't cascade. For end-to-end processes, focus on the highest-risk steps first.
Can I combine probabilities from different sources?
Only if the events are truly independent and the probability estimates are for the same time period and conditions. Mixing data from different suppliers, time periods, or process configurations can lead to incorrect results. Always document assumptions about independence and data sources.