Probability is the language of uncertainty, and a probability calculator is one of the fastest ways to speak it. When you know the chance of two separate events, this tool instantly works out everything that can happen between them: both occurring together, at least one occurring, exactly one occurring, or neither occurring at all. Just enter the probability of event A and the probability of event B as percentages, and the calculator returns the union, the intersection, the complements and the exclusive-or in a single view.
How to Calculate the Probability of Events
Probability measures how likely an event is, expressed as a number between 0 and 1 or as a percentage between 0 and 100. A probability of 0 means the event is impossible; a probability of 1 means it is certain. In everyday terms, probability is the number of favorable outcomes divided by the total number of possible outcomes, but when you already know P(A) and P(B), the rest of the work is algebra built on a few simple rules.
The calculator takes those two known probabilities and derives five more: the chance A does not happen, the chance B does not happen, the chance both happen, the chance at least one happens, and the chance exactly one happens. Every one of these follows from the same two inputs, so the results form one consistent picture rather than five separate guesses.
The Complement: P(A') and P(B')
The complement is the easiest probability to find and the foundation for several others. If the probability that an event happens is P(A), then the probability that it does not happen, written P(A'), is simply one minus P(A). Because every outcome either satisfies the event or does not, the two probabilities always add up to exactly 100 percent. If there is a 15 percent chance that a package is delivered late, then there is an 85 percent chance it arrives on time. This calculator shows both complements instantly, so you can always see the full picture on either side of an event.
The Intersection: P(A and B)
The intersection is the probability that both events occur together, written P(A and B). For independent events, this is a simple multiplication: multiply the two individual probabilities. If event A has a 60 percent chance and event B has a 40 percent chance, the probability that both happen is 60 percent of 40 percent, which is 24 percent. The calculator uses exactly this product rule, and it is the reason the two-event model is so tidy. When you roll two separate dice, for example, the chance of rolling a six on the first is one in six and the chance of rolling a six on the second is also one in six, so the chance of getting two sixes is one in thirty-six.
The Union: P(A or B)
The union is the probability that at least one of the two events occurs, written P(A or B). To compute it you add the two individual probabilities and then subtract the intersection, because the outcome where both happen belongs to both events and must not be counted twice. With a 60 percent chance of A and a 40 percent chance of B, adding them gives 100 percent, but subtracting the 24 percent overlap brings the union down to 76 percent. That 76 percent is the chance that A, or B, or both, will occur, and it is the figure most people actually mean when they ask how likely "either one" is.
The Exclusive-Or: P(A or B, but Not Both)
Sometimes the question is about exactly one outcome, not at least one. The exclusive-or, written P(A XOR B), counts the situations where A happens without B, or B happens without A, but never both together. The calculation adds P(A) and P(B) and subtracts twice the intersection, because the overlapping outcome must be removed from both sides. For the running example, that is 60 plus 40 minus twice 24, which equals 52 percent. The exclusive-or is the probability of a strictly one-sided result, and it is the number to use when the joint outcome is not acceptable.
The Probability That Neither Event Occurs
The last derived result is the chance that both events fail, often phrased as the probability of neither A nor B. This is simply the complement of the union: whatever the union covers, everything else is the neither case. Subtract each individual probability from 100 percent and add back the intersection, or more directly, take 100 percent minus the union. In the example above, that is 24 percent. The four outcomes of the donut chart neatly summarize the entire space: A only, B only, both, and neither, and these four slices always total exactly 100 percent.
Independence Is the Key Assumption
All of the formulas above rely on one assumption: the two events are independent. Independence means that knowing one event occurred tells you nothing about the other. Two separate coin flips are independent because the first flip cannot influence the second. By contrast, drawing two cards from a deck without replacement is dependent, because removing the first card changes the odds for the second. When events are dependent, the intersection is no longer a simple product, and you need conditional probability instead. This calculator assumes independence, so it is the right tool for independent trials, separate systems, and unrelated forecasts.
A Worked Example with Dice
To see the rules in action, imagine two independent rolls of a fair six-sided die. Let event A be rolling a number greater than two, which happens four times out of six, so P(A) is about 66.7 percent. Let event B be rolling an even number, which happens three times out of six, so P(B) is 50 percent. Entering these into the calculator gives a 33.3 percent chance that both conditions hold, a roll that is above two and even. The union comes to about 83.3 percent, meaning most rolls satisfy at least one of the conditions. The exclusive-or lands at about 50 percent, and the probability of neither, a roll of exactly two, is 16.7 percent. Each of these answers can be checked by hand by listing the six outcomes, which is a satisfying confirmation that the formulas are faithful.
Reading the Probability Space Donut
The donut chart divides the entire sample space into four mutually exclusive regions: outcomes where only A happens, outcomes where only B happens, outcomes where both happen, and outcomes where neither happens. Because the regions do not overlap, their percentages always add to 100. The center of the donut highlights the joint probability, the chance that both events occur, while the note beneath calls out the union. This visual layout makes it easy to see at a glance how much of the space is covered by the events and how much lies outside them.
Entering Percentages Versus Fractions
The calculator asks for both probabilities as whole percentages, which matches how people naturally talk about chance. The formulas, however, work with fractions, so the calculator converts internally by dividing by 100 and then converts the results back to percentages for display. Treating a 24 percent result as the decimal 0.24 is the single most common error in hand calculations, and it is entirely handled here. Every output is already a percentage, so 24 means 24 percent, ready to compare against any other risk figure you care about.
Entering a probability larger than 100 or smaller than 0 is physically impossible for an event that may or may not happen, so the sliders keep the inputs in the valid range. If your situation is better described as odds, for example five to one, convert the odds to a probability first by dividing the favorable count by the total number of outcomes, then enter that value into the calculator.
Common Mistakes When Working with Probabilities
- Treating a percentage as a fraction in the middle of a calculation and misplacing the decimal point, for example multiplying 0.60 by 40 instead of 0.40.
- Forgetting to subtract the overlap when computing the union, which double-counts the outcomes where both events occur and inflates the result.
- Assuming independence without checking, then multiplying probabilities for events that clearly influence each other, such as weather and outdoor plans.
- Confusing the union, at least one, with the exclusive-or, exactly one; the exclusive-or drops the joint outcome entirely.
- Expecting P(A) and P(B) to sum to 100 percent, which is only true when the events are complements of each other.
- Misreading the neither case as simply 100 minus P(A) and ignoring the overlap that has to be added back.
Key Assumptions
- The two events are independent, so the joint probability equals the product of the individual probabilities.
- Both inputs are valid probabilities between 0 and 100 percent, and results are only meaningful within that range.
- The events are not mutually exclusive, meaning they may occur together, and the overlap is handled by subtraction.
- All derived probabilities are rounded for display but computed at full precision, so they remain consistent with one another.
From dice rolls to risk assessments to quality control, the probability of two independent events is one of the most frequently needed calculations in statistics. Enter P(A) and P(B) and this calculator gives you the complete relationship between them: the intersection, the union, the complements, the exclusive-or and the neither case, all on one page.
Disclaimer
Results are provided as estimates for informational purposes only and may be inaccurate. Always verify outcomes with a qualified professional before making financial or personal decisions based on these calculations.