Probability Calculator
Results
| Occurs — P(A) | — |
|---|---|
| Does Not Occur — P(A') | — |
Results
| Both Occur — P(A ∩ B) | — |
|---|---|
| At Least One Occurs — P(A ∪ B) | — |
| Exactly One Occurs — P(A ⊕ B) | — |
| Neither Occurs | — |
Results
| Exactly k successes — P(X = k) | — |
|---|---|
| At least k successes — P(X ≥ k) | — |
| At most k successes — P(X ≤ k) | — |
| Less than k successes — P(X < k) | — |
| More than k successes — P(X > k) | — |
How to Use This Probability Calculator
Probability homework jumps between three different setups—one event, two independent events, and binomial trials—and each setup uses a different formula sheet row. Keeping complement, union, and "exactly k successes" straight on a timed quiz is where most students I tutor lose marks. This calculator splits those three jobs into tabs so you enter numbers once and read every related probability in one table.
- Single Event tab. Enter P(A) as a decimal (0.4) or percentage (40%). Results show P(A) and its complement P(A').
- Two Events tab. Enter P(A) and P(B); events are treated as independent. You get intersection, union, exactly-one, and neither probabilities.
- Binomial tab. Enter trials n, success probability p, and target successes k. The table lists exact, at-least, at-most, less-than, and greater-than counts.
- Cross-check counts. Binomial formulas use combinations—verify C(n,k) with the Permutation and Combination Calculator when your instructor wants the counting step shown separately.
Probability Formulas and Practical Applications
Probability measures how likely an outcome is on a scale from impossible (0) to certain (1). Think of it as a fraction of favorable outcomes over all outcomes—except many real problems encode the fraction directly as P(A).
Complement rule
P(A') = 1 − P(A)
"At least one" problems often flip to 1 − P(none) instead of adding many cases—quality control and medical testing use this constantly.
Two independent events
P(A ∩ B) = P(A) × P(B)
P(A ∪ B) = P(A) + P(B) − P(A) × P(B)
Intersection is "both happen"; union is "at least one happens." Exactly-one and neither rows expand those building blocks for coin-and-die style problems.
Binomial probability
P(X = k) = C(n,k) × p^k × (1 − p)^(n − k)
Fixed n trials, constant success chance p, independent trials—coin flips, defect rates, and multiple-choice guessing fit this model. Summing P(X = k) for a range gives "at least" or "at most" answers shown in the binomial tab.
Standard Units and Conversion Tables
Probabilities are unitless but may be expressed as decimals or percentages:
| Decimal | Percent | Meaning |
|---|---|---|
| 0 | 0% | Impossible |
| 0.25 | 25% | One in four |
| 0.5 | 50% | Even chance |
| 1 | 100% | Certain |
For random sampling without replacement, counts differ—see the Random Number Generator for draws, or browse all math calculators.
Frequently Asked Questions
What are independent events?
One outcome does not shift the other's chance. Rolling a die and flipping a coin are the classic pair—the Two Events tab assumes independence.
Can a probability be negative or greater than 100%?
No. Valid inputs stay between 0 and 1 (or 0% and 100%). Out-of-range values trigger an error instead of a result.
How does the binomial tab work?
Enter n, p, and k. The tool applies the binomial formula and sums terms for cumulative rows like P(X ≥ k).
Can I enter probabilities as percentages?
Yes—type 40% or 0.4; both normalize before calculation.
Does the two-events tab assume dependence?
No. It uses independent-event formulas only. Dependent cases need conditional probability, which is outside this tool's scope.
Disclaimer. RapidRatio is informational only—not statistical or professional advice. Verify models and inputs with a qualified instructor before high-stakes decisions.