Independent Events Calculator

Analyze intersections, unions, complements, and outcome cases. Switch between two or three events with ease. See charts, trial estimates, and downloadable summaries for decisions.

Compute intersections, unions, complements, exact outcome cases, expected frequencies, and visual summaries for two or three independent events.

Calculator Inputs

Use the responsive form below. It shows three columns on large screens, two on smaller screens, and one on mobile.

Choose whether to analyze two or three independent events.
The calculator converts percentages into probabilities automatically.
Used for expected counts beside each probability result.
Use a short event name, such as A or Rain.
Use another short event name for the second event.
Shown only for the three-event mode.
Enter a decimal or percent based on your chosen unit.
Independent events multiply for joint probabilities.
Needed only when three events are active.
Use this to compare an observed joint value against the theoretical model.

Example Data Table

These sample scenarios show how independent event outcomes change with different inputs.

Scenario Input Values Key Result Interpretation
Two Events P(A)=0.55, P(B)=0.40, Trials=1000 P(A ∩ B)=0.2200, P(A ∪ B)=0.7300 About 220 joint cases and 730 cases with at least one event.
Three Events P(A)=0.55, P(B)=0.40, P(C)=0.30, Trials=1000 P(A ∩ B ∩ C)=0.0660, P(at least one)=0.8740 About 66 triple cases and 874 trials with at least one event.
Balanced Inputs P(A)=0.50, P(B)=0.50, Trials=500 P(exactly one)=0.5000 Half the trials should show exactly one event happening.

Formula Used

For independent events, intersections are products because one event does not change the probability of another.

Two-event formulas

P(A ∩ B) = P(A) × P(B)

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

P(only A) = P(A) × (1 - P(B))

P(only B) = (1 - P(A)) × P(B)

P(exactly one) = P(only A) + P(only B)

P(neither) = (1 - P(A)) × (1 - P(B))

P(A | B) = P(A) and P(B | A) = P(B)

Three-event formulas

P(A ∩ B ∩ C) = P(A) × P(B) × P(C)

P(no event) = (1 - P(A)) × (1 - P(B)) × (1 - P(C))

P(at least one) = 1 - P(no event)

P(only A) = P(A) × (1 - P(B)) × (1 - P(C))

P(exactly two) = P(A)P(B)(1-P(C)) + P(A)P(C)(1-P(B)) + P(B)P(C)(1-P(A))

Expected cases = Probability × Number of trials

How to Use This Calculator

Follow these steps to get clean and meaningful probability results.

  1. Choose whether your scenario has two events or three.
  2. Select decimal or percent input mode.
  3. Enter labels for your events, if you want descriptive names.
  4. Type the probability of each event.
  5. Add the number of trials to estimate expected frequencies.
  6. Optionally enter an observed joint or triple value for comparison.
  7. Press the calculate button to show results above the form.
  8. Review the table, graph, and export buttons for reporting.

FAQs

1) What are independent events?

Independent events are events where one outcome does not change the chance of the other. Their joint probability is found by multiplying their individual probabilities.

2) Why does the calculator multiply probabilities?

For independent events, the multiplication rule applies directly. That is why the intersection of two or three independent events is the product of their separate probabilities.

3) Can I enter percentages instead of decimals?

Yes. Change the input unit to percent, then enter values such as 55 or 40. The calculator converts them into decimal probabilities automatically.

4) What does exactly one event mean?

Exactly one event means one chosen event happens while all others do not. In a two-event case, it is the sum of only A and only B.

5) What does the expected cases column show?

It estimates how many times each result should occur across your trial count. The value is simply the probability multiplied by the number of trials.

6) Why compare observed and theoretical joint values?

That comparison helps you see whether your measured data fits the independence assumption. Large differences suggest the events may not behave independently in practice.

7) Does the union mean one event or both events?

Yes. The union represents at least one event happening. It includes only A, only B, and both together for the two-event case.

8) When should I use the three-event mode?

Use it when your problem involves three separate independent events. It adds pair intersections, triple intersection, exactly two events, exactly one event, and none.

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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.