What Is Independent Event Probability?
Two events are independent when one does not change the probability of the other. For independent events the probability that all occur is the product of their probabilities: P(A and B) = P(A) × P(B).
Enter the probability of each event as decimals or percentages. The calculator returns the probability that all events occur, that none occur, that at least one occurs, that exactly one occurs and, optionally, exactly k, together with the expected count and a chart of the full distribution of how many events occur.
The article below explains the formulas, variables, a worked three-event example and answers about the multiplication rule and the "at least one" shortcut.
How to Use the Independent Event Probability Calculator
- Enter the probability of each event, separated by commas.
- Choose decimals or percentages.
- Optionally enter k for the chance that exactly k events occur.
- Read P(all), P(at least one), P(none) and the full distribution.
Independent Event Probability Formula
All events occur
At least one occurs
Expected count
Independent Event Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| pᵢ | Probability of event i | |
| Π pᵢ | Product of probabilities | P(all events occur) |
| Π(1 − pᵢ) | Probability that none occur | |
| k | Number of events that occur |
Independent Event Probability Worked Example
Three independent events have probabilities 0.5, 0.3 and 0.2. All three occur with probability 0.5 × 0.3 × 0.2 = 0.03. None occurs with probability 0.5 × 0.7 × 0.8 = 0.28, so at least one occurs with probability 1 − 0.28 = 0.72.
With the example values loaded in the calculator above, the steps are:
- Independent events multiply: P(all) = 0.5 × 0.3 × 0.2 × 0.6 = 0.018.
- P(none) = (1 − 0.5) × (1 − 0.3) × (1 − 0.2) × (1 − 0.6) = 0.112.
- P(at least one) = 1 − P(none) = 1 − 0.112 = 0.888.
- The distribution of the number of events that occur is built one event at a time (a Poisson-binomial distribution).
Result: P(all events occur) = 0.018; P(at least one occurs) = 0.888; P(exactly 2 occur) = 0.37.
How to Read Independent Event Probabilities
Events are independent when one happening does not change the chance of another. Then "all occur" is the product of the individual probabilities, and "none occur" is the product of the complements. For probabilities 0.5, 0.3 and 0.2, all three occur with probability 0.03, and none occur with probability 0.28.
"At least one" is 1 minus "none", here 0.72. The expected number of events is the sum of the probabilities, 1.0, and the table shows how the chance is spread over 0, 1, 2 and 3 events.
Where Independent Event Probability Is Used
Reliability
Find the chance that every part of a system works, or that at least one backup does, when failures are independent.
Games and cards
Work out the chance of several separate trials all going your way, such as three coin flips or three separate draws with replacement.
Risk planning
Combine small independent risks into the chance that at least one occurs over a year.
Homework
Check "all", "none", "at least one" and "exactly k" answers against the full distribution table.
Common Independent Event Probability Mistakes to Avoid
- Multiplying is only valid for independent events. Drawing cards without replacement makes the events dependent.
- Use "none occur" and subtract from 1 for any "at least one" question. It is shorter and less error-prone.
- Enter probabilities as decimals or percentages, but not a mix. The calculator asks which form you are using.
- Small probabilities do not add up exactly. Three 10% events give 27.1% for at least one, not 30%.
Independent Event Probability Terms and Definitions
- Independent events
- Events where the occurrence of one does not change the probability of another.
- Poisson-binomial
- The distribution of the number of successes when trials have different probabilities.
Independent Event Probability FAQs
How do you find the probability of independent events?
Multiply the individual probabilities for "all occur". For probabilities 0.5, 0.3 and 0.2 that is 0.5 × 0.3 × 0.2 = 0.03. The calculator also gives "none", "exactly k" and "at least k".
How do I find the probability of at least one event?
Compute 1 − P(none). With the same three events, P(none) = 0.5 × 0.7 × 0.8 = 0.28, so P(at least one) = 0.72.
What is the probability of A or B for independent events?
Use P(A) + P(B) − P(A)·P(B). For 0.5 and 0.3 that is 0.5 + 0.3 − 0.15 = 0.65. Simply adding would give 0.8 and double-count the overlap.
How can I tell if events are independent?
Events are independent if P(A and B) = P(A) × P(B), or if knowing one happened does not change the chance of the other. Separate coin flips are independent. Draws without replacement are not.