Probability
Probability Calculator
Apply one probability rule at a time: a complement, a union with the overlap removed, or at least one success in n independent trials.
Worksheet
Press Calculate to see the result.
Apply one probability rule at a time: a complement, a union with the overlap removed, or at least one success in n independent trials.
What this calculator answers
Many probability questions are one algebraic rule, not a named distribution. The chance that an event fails to happen is one minus its probability. The chance that at least one of two events happens adds the two probabilities and then subtracts the overlap so it is not counted twice. The chance of at least one success in a run of independent trials is one minus the chance that every trial fails. This page keeps those three rules separate so a complement is never mixed up with a union, and a union is never computed with a pooled proportion from a hypothesis test.
Complement returns 1 − p. Union returns P(A) + P(B) − P(A and B), after checking that the three numbers can belong to a pair of events. At least one returns 1 − (1 − p)^n for independent trials with the same p. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Complement, union, and at least one success uses this relationship.
Complement: 1 − p. Union: P(A) + P(B) − P(A and B). At least one: 1 − (1 − p)^n.
- p. A single event probability, or the success probability on one trial.
- n. How many independent trials are repeated.
- P(A and B). The overlap removed from a union.
- 1 − p. The probability that the event does not happen.
The complement is the rest of a probability that totals 1. The union inclusion is an addition that double-counts the overlap, so the joint is subtracted once. Independence across trials turns n failures into a product (1 − p)^n, and the complement of that product is the event that the failures were not universal. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The selected rule is the complement, and p is 0.25. The complement is 1 − 0.25 = 0.75. This is the probability that the event does not happen.
The union fields were not used. A union would subtract an overlap. The trial count was not used. At least one success would raise a failure probability to the power n. p = 0.25 sits between 0 and 1, so the complement does too.
Adding the event and its complement returns 1: 0.25 + 0.75 = 1. A percent entry such as 25 would have been rejected. The box wants the proportion 0.25.
| Complement 1 − p | 0.75 |
|---|---|
| p | 0.25 |
| Rule | Complement |
A second situation
The union rule with the overlap removed
The selected rule is the union. P(A) is 0.4 and P(B) is 0.5. The overlap P(A and B) is 0.1. Adding the two probabilities gives 0.9, which counts the overlap twice.
Subtract the overlap once: 0.9 − 0.1 = 0.8. The complement rule was not used. It would have needed only one probability. The joint 0.1 does not exceed either event probability.
The union 0.8 lies between 0 and 1. This is not a pooled proportion and not a hypothesis test. It is the inclusion rule for two events.
| P(A or B) | 0.8 |
|---|---|
| P(A) + P(B) | 0.9 |
| Overlap removed | 0.1 |
| Rule | Union |
Switching from the complement of 0.25 to the union of 0.40 and 0.50 with overlap 0.10 changes both the inputs and the rule. The union is 0.80 only after the overlap is removed. Adding 0.40 and 0.50 without that subtraction would claim 0.90 and count the overlap twice. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
p moved from 0.25 to 0.10, so the complement rose. The headline result went from 0.75 to 0.9.
The nudged inputs
The selected rule is the complement, and p is 0.1. The complement is 1 − 0.1 = 0.9. This is the probability that the event does not happen.
The union fields were not used. A union would subtract an overlap. The trial count was not used. At least one success would raise a failure probability to the power n. p = 0.1 sits between 0 and 1, so the complement does too.
Adding the event and its complement returns 1: 0.1 + 0.9 = 1. A percent entry such as 25 would have been rejected. The box wants the proportion 0.1.
| Complement 1 − p | 0.9 |
|---|---|
| p | 0.1 |
| Rule | Complement |
How to read the result
Read the headline as the probability of the event the selected rule describes. A complement near 1 means the original event was rare. A union is not a sum of the two inputs. An at-least-one result near 1 means repeated trials made a miss on every trial unlikely. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Every typed probability is between 0 and 1.
- The union's joint does not exceed either event, and the union itself does not exceed 1.
- At least one assumes independent trials with a shared p.
- n is a whole number of trials, at least 1.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Adding P(A) and P(B) for a union and forgetting to subtract the overlap.
- Using 1 − p^n for at least one success. The power belongs on the failure probability, not on p.
- Treating dependent trials as independent. The product (1 − p)^n is then the wrong model.
- Entering a percent. The boxes want proportions such as 0.25.
Where a different calculator is the honest choice
Use the conditional probability calculator when the question is A given B. Use the Bayes theorem calculator for a screening tree. Use the binomial calculator when you need the probability of an exact count, not merely the probability of one or more successes. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Conditional Probability Calculator, Bayes Theorem Calculator, Binomial Distribution Calculator.
Check yourself
What is the complement of p = 0.25?
The complement is 1 − 0.25 = 0.75. That is the probability the event does not happen.
Using the result in a sentence
The rest of a known risk, the chance that either of two overlapping events happens, and the chance that a repeated independent trial eventually succeeds are the three jobs on this page. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Three rules cannot cover every probability story. Dependence, more than two events in a union, and unequal trial probabilities need a different worksheet. The at-least-one shortcut also hides the individual counts that a binomial mass would show. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Probability Calculator commit to?
This page uses Complement, union, and at least one success: Complement: 1 − p. Union: P(A) + P(B) − P(A and B). At least one: 1 − (1 − p)^n.. The complement is the rest of a probability that totals 1. The union inclusion is an addition that double-counts the overlap, so the joint is subtracted once. Independence across trials turns n failures into a product (1 − p)^n, and the complement of that product is the event that the failures were not universal. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Rule, Probability p, Trials n, P(A), P(B), P(A and B), and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the complement 1 − p is 0.75. Read the headline as the probability of the event the selected rule describes. A complement near 1 means the original event was rare. A union is not a sum of the two inputs. An at-least-one result near 1 means repeated trials made a miss on every trial unlikely. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Adding P(A) and P(B) for a union and forgetting to subtract the overlap. Three rules cannot cover every probability story. Dependence, more than two events in a union, and unequal trial probabilities need a different worksheet. The at-least-one shortcut also hides the individual counts that a binomial mass would show. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the conditional probability calculator when the question is A given B. Use the Bayes theorem calculator for a screening tree. Use the binomial calculator when you need the probability of an exact count, not merely the probability of one or more successes. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The complement is 1 − 0.25 = 0.75. That is the probability the event does not happen. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.