Probability Calculator Examples With Steps
Follow independent, mutually exclusive, overlapping, conditional, and binomial probability examples with exact and cumulative results.
How should you use these Probability Calculator examples?
These examples turn the Probability Calculator inputs into complete scenarios with stated values, calculation steps, and interpreted results. Use them to check how the calculator behaves, compare your own situation with realistic cases, and identify which inputs have the greatest effect before relying on an estimate.
Independent Events Example
If P(A) = 40% and P(B) = 30%, then P(A and B) = 0.40 × 0.30 = 12%. The probability of A or B is 40% + 30% - 12% = 58%. Neither occurs with probability 42%.
Mutually Exclusive Example
If P(A) = 20% and P(B) = 35% and the events cannot occur together, their intersection is 0%. The union is 55%, and neither occurs with probability 45%.
Known Overlap Example
Let P(A) = 60%, P(B) = 50%, and P(A and B) = 30%. Then A only is 30%, B only is 20%, either is 80%, and neither is 20%. P(A given B) is 30% ÷ 50% = 60%.
Exactly Three Successes
For 10 independent trials with p = 50%, P(X = 3) = C(10,3)(0.5)³(0.5)⁷ = 11.72%.
At Most and At Least
For the same 10 trials, P(X at most 3) is 17.19% after adding outcomes 0 through 3. P(X at least 3) is 94.53% after adding outcomes 3 through 10.
Edge Cases
When p = 0, zero successes has probability 100%. When p = 1, n successes has probability 100%. A conditional probability is undefined when its conditioning event has probability zero.
What the Examples Show
For “What the Examples Show,” the Probability Calculator examples organizes the relevant inputs, applies the documented method, and shows a result that can be compared across scenarios. The points below identify the specific considerations and assumptions users should review before relying on the estimate.
- Independent and mutually exclusive relationships use different intersections.
- A known overlap supports conditional probability calculations.
- Exactly k uses one binomial term, while at most and at least sum several terms.
- Boundary probabilities concentrate all outcomes at zero or n successes.
Try each scenario in the Probability Calculator and compare the diagram or distribution table with the numeric results.
Corrections and Review
Report an issue or error in the calculator using the Contact Us page.