Probability Calculator — Odds for Single & Two Events
How likely is it, really? Count favourable outcomes for a single event, or combine two independent events — both, either, exactly one, neither — and see every answer as a percentage, a fraction and odds.
Your events
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The odds
The picture — a Venn diagram
Each circle is one event; the overlap is where both happen, and everything outside is “neither”.
- Two independent events at 30% and 40%: P(A and B) = 12.00%, P(A or B) = 58.00% and P(neither) = 42.00% — every region of the Venn diagram, as percentages, fractions and odds.
- A single event is a simple count: 3 favourable outcomes out of 12 totals 25.00% — also shown as the fraction 1/4 and odds 1:3.
- A 10% chance tried 10 times lands at least once 65.13% of the time, and on the standard bell curve P(X ≤ 1) is 84.13% — the two extra tools below the main calculator.
- Two events — solve from any two values
- Repeated trials — will it happen at least once?
- The normal distribution — the bell curve
- Bayes’ theorem — what a positive result really means
- Draws without replacement — the hypergeometric case
- Keep exploring
- How it works
- The math behind the odds
- Go deeper
- Frequently asked questions
- Related calculators
Two events — solve from any two values
The tool at the top starts from P(A) and P(B). This one starts from whichever two values you actually know — the overlap, the union, even P(A | B) — and works out the rest. If the events influence each other, switch to dependent mode and hand it P(A | B) directly.
What do you know?
The full set
Repeated trials — will it happen at least once?
The same chance, tried again and again: ten lottery tickets, five coin flips, a week of rain risk. “At least once” climbs fast — this is the complement rule doing the heavy lifting.
The trials
The repeated odds
The full distribution — every possible count
Not just one k: the chance of every success count these trials could produce, with the running totals that answer “at most” and “at least” questions.
| Successes k | P(X = k) | P(X ≤ k) | P(X ≥ k) |
|---|---|---|---|
| 0 | 34.8678% | 34.8678% | 100.0000% |
| 1 | 38.7420% | 73.6099% | 65.1322% |
| 2 | 19.3710% | 92.9809% | 26.3901% |
| 3 | 5.7396% | 98.7205% | 7.0191% |
| 4 | 1.1160% | 99.8365% | 1.2795% |
| 5 | 0.1488% | 99.9853% | 0.1635% |
| 6 | 0.0138% | 99.9991% | 0.0147% |
| 7 | 0.0009% | 100.0000% | 0.0009% |
| 8 | 0.0000% | 100.0000% | 0.0000% |
| 9 | 0.0000% | 100.0000% | 0.0000% |
| 10 | 0.0000% | 100.0000% | 0.0000% |
The normal distribution — the bell curve
Heights, test scores, measurement errors: when values cluster around an average, the bell curve gives the chance of landing at or below any value — no outcome counting required.
The curve
The probability
Work backwards — from a probability to the value
The inverse question: which value leaves a given share of the curve below it? It uses the mean and standard deviation from the curve above.
The percentile
The value
Confidence-interval table
For the curve above — mean and standard deviation as entered — the central range that holds each standard share of the values:
| Share of values | z | From (μ − zσ) | To (μ + zσ) |
|---|---|---|---|
| 50% | 0.674 | −0.67449 | 0.67449 |
| 68% | 0.994 | −0.994458 | 0.994458 |
| 80% | 1.282 | −1.28155 | 1.28155 |
| 90% | 1.645 | −1.64485 | 1.64485 |
| 95% | 1.960 | −1.95996 | 1.95996 |
| 99% | 2.576 | −2.57583 | 2.57583 |
| 99.9% | 3.291 | −3.29053 | 3.29053 |
Bayes’ theorem — what a positive result really means
A test can be 99% sensitive and still be wrong most of the time — when the condition is rare, the false positives outnumber the true ones. Bayes’ theorem is the arithmetic that weighs a result against the base rate.
The test and the base rate
Behind a positive result
Draws without replacement — the hypergeometric case
Cards from a deck, marbles from a bag, inspectors sampling a batch: when nothing goes back, the deck shrinks and the simple multiply rule no longer applies. This is the distribution that handles it exactly.
The draw
The draw odds
Keep exploring
Three companion tools for data and averages.
How it works
Count, divide, combine — that is all probability ever does.
Pick your mode
One event on its own, or two independent events combined — the inputs and the picture change to match.
Enter the chances
Outcomes for a single event, or a percentage for each of two events. Independence is assumed in two-event mode.
Read the full breakdown
Not just the headline: both, either, exactly one and neither — or the fraction, decimal and odds — so the whole picture adds up to 100%.
The math behind the odds
Every probability is a share of the possible: favourable outcomes divided by total outcomes. Combining events follows two rules. For independent events, “and” means multiply — P(A and B) = 30% × 40% = 12.00%. “Or” means add, then subtract the overlap you counted twice — P(A or B) = 30% + 40% − 12% = 58.00%. The rest of the breakdown is bookkeeping: A only is 30% − 12% = 18.00%, B only is 40% − 12% = 28.00%, exactly one is 18% + 28% = 46.00%, and neither is the complement, 100% − 58% = 42.00%. All four regions sum to exactly 100% — check them on the Venn diagram above.
Worked examples:
| Question | Working | Result |
|---|---|---|
| At least one of A (30%) or B (40%)? | 30 + 40 − (30 × 40 ÷ 100) = 70 − 12 | 58.00% |
| 3 favourable outcomes out of 12 | 3 ÷ 12 = 1/4 · odds in favour 3 : 9 = 1 : 3 | 25.00% |
| Rolling a six on one die | 1 favourable face out of 6 | 16.67% |
| Neither A nor B | (100 − 30) × (100 − 40) ÷ 100 = 70% × 60% | 42.00% |
The sanity checks: every probability lives between 0% (impossible) and 100% (certain), and the regions of any breakdown must sum to 100% — here, 18 + 12 + 28 + 42 = 100. If your regions do not add up, the overlap was double-counted or the complement was forgotten; those two slips account for nearly every wrong probability answer.
Go deeper
The two distinctions that decide whether your answer is right.
These sound similar and behave oppositely:
- Independent: one event does not change the other’s odds — coin flips, dice rolls, two separate draws with replacement. They CAN happen together, and P(and) = multiply.
- Mutually exclusive: the events cannot happen together at all — one card cannot be both a king and a queen. P(and) = 0, so P(or) is a clean add with nothing to subtract.
- The trap: treating exclusive events as independent invents an overlap that does not exist; treating independent events as exclusive erases one that does.
- This calculator uses the independent model in two-event mode — right for flips, rolls, weather-style estimates and most textbook questions.
The same chance wears two notations:
- Probability compares success with everything: 3 wins out of 12 outcomes is 3/12 = 25.00%.
- Odds compare success with failure: 3 wins against 9 losses is odds of 1 : 3 in favour (and 3 : 1 against).
- Converting: odds of a : b in favour mean a probability of a ÷ (a + b) — 1 : 3 gives 1 ÷ 4 = 25%. Probability p gives odds of p : (1 − p).
- Where you meet odds: bookmakers and board games quote odds; science and statistics quote probability. Being bilingual between them prevents expensive misunderstandings.
Frequently asked questions
Quick answers about probability.
How do I calculate probability?
Divide the favourable outcomes by the total equally likely outcomes: P = favourable ÷ total. Three winning outcomes out of 12 gives 3 ÷ 12 = 0.25 = 25.00%. Multiply by 100 for the percentage, and simplify the fraction (3/12 = 1/4) for the tidiest form. The single-event mode of this calculator does all three, plus the odds.
How do I find the probability of two events both happening?
If the events are independent, multiply: P(A and B) = P(A) × P(B). With P(A) = 30% and P(B) = 40%: 0.30 × 0.40 = 0.12, so 12.00%. “And” shrinks the probability — both things must go right at once — which is why joint chances are always smaller than either event alone.
How do I find the probability that at least one event happens?
Add the two probabilities, then subtract the overlap you counted twice: P(A or B) = P(A) + P(B) − P(A and B) = 30% + 40% − 12% = 58.00%. The mirror-image trick is just as useful: P(at least one) = 100% − P(neither) = 100% − 42% = 58%. Both routes agree — they must.
What does it mean for events to be independent?
Knowing one happened tells you nothing about the other: a coin does not remember its last flip, and a die does not lean toward numbers it has skipped. Contrast drawing cards without replacement — each draw changes what is left in the deck, so those events are dependent and the simple multiply rule needs adjusting. This calculator’s two-event mode assumes independence — unless you switch the relationship setting to mutually exclusive, for events that can never happen together. For dependent events that can still happen together, use the two-event solver on this page and enter P(A | B) directly.
What is the difference between probability and odds?
Probability compares successes with all outcomes; odds compare successes with failures. A 25.00% probability (1 in 4) is odds of 1 : 3 in favour — one win for every three losses. Convert odds a : b back with a ÷ (a + b). Bookmakers quote odds; scientists quote probability; they are the same fact in two dialects.
What is the complement of an event?
Everything the event is not. P(complement) = 100% − P(event), so a 25.00% chance has a 75.00% complement. Complements are the master key to “at least one” questions: instead of adding up many winning cases, compute the single losing case — P(at least one) = 100% − P(none). For the default events, P(neither) = 42.00%, so P(at least one) = 58.00%.
Related calculators
More tools for data, spread and averages.
Disclaimer
This calculator is for learning and quick reference. Single-event mode assumes every outcome is equally likely, and two-event mode follows the relationship you pick for the events — independent or mutually exclusive; real-world events (weather, health, markets) are often neither, so treat the results as exact math on stated assumptions, not as predictions.