Do all probabilities have to add to one?

The probabilities of a complete set of mutually exclusive alternatives do. An arbitrary list of overlapping events need not total one.

Are the axioms a method for estimating probabilities?

No. They specify consistency rules; estimation requires assumptions and evidence.

Does a probability of zero always mean impossible?

Not in continuous models, where individual outcomes can have probability zero even though some individual outcome occurs.

AI Glossary / Graphs and symbolic AI / Beginner

Kolmogorov's axioms

Definition

Kolmogorov's axioms are three basic rules that define a probability measure: event probabilities are nonnegative, the probability of the whole sample space is one, and probabilities add over any countable collection of mutually exclusive events. Together, they provide a consistent mathematical foundation for calculating with uncertainty.

What it means

The sample space contains the possible outcomes under a chosen model. An event is an allowed set of those outcomes. The axioms constrain the numbers assigned to events; they do not by themselves tell us which model accurately describes the world or how to estimate its probabilities.
For a finite example, imagine a package arriving early, on time, or late, with those categories defined to be mutually exclusive and exhaustive. Their probabilities cannot be negative and must add to one. More general probability models also need a suitable collection of measurable events, which makes probability assignments well defined in settings with infinitely many possible outcomes.

How it works

The first axiom requires P(A) to be at least zero for every event A. The second requires P(S) = 1, where S is the sample space. The third says that the probability of the union of a countable sequence of pairwise disjoint events equals the sum of their probabilities.
From these rules, one derives familiar consequences: probabilities cannot exceed one, the empty event has probability zero, and an event's complement has probability one minus the event's probability. For overlapping events, simple addition double-counts the overlap, so P(A or B) = P(A) + P(B) - P(A and B). Independence is a separate relationship, not an automatic consequence of the axioms.

Three practical examples

A forecast dashboard checks that mutually exclusive delivery categories have nonnegative probabilities totaling 100 percent.
A simulation combines several disjoint failure modes without counting the same modeled outcome twice.
An analyst checks an “either issue occurs” estimate by identifying whether the two issue categories can overlap.

Think of it this way

Think of allocating one whole unit of probability across an event model. Disjoint portions can be added, but overlapping portions cannot be counted twice. The analogy is limited because probability is not always distributed across finitely many individual outcomes with positive shares; continuous models can assign probability zero to each individual point while assigning positive probability to intervals.

Worked example

Consider an invented model for a service request. Let A mean that a request needs translation, and B mean that it needs an attachment review. Suppose P(A) = 0.30, P(B) = 0.25, and P(A and B) = 0.10.
Then P(A or B) = 0.30 + 0.25 - 0.10 = 0.45. Adding 0.30 and 0.25 without subtracting the overlap would give 0.55 and count requests needing both services twice. The probability that neither service is needed is 1 - 0.45 = 0.55.
We can check the model using four mutually exclusive categories. Translation only has probability 0.20; attachment review only has probability 0.15; both has probability 0.10; and neither has probability 0.55. These four nonnegative numbers total 1.00.
The model does not imply independence. If A and B were independent, their joint probability would be 0.30 times 0.25 = 0.075, which differs from the stipulated 0.10. The example therefore obeys the probability axioms while describing dependent events. All numbers are invented to illustrate internal consistency, not estimated from actual service data.

Illustrative case study

In a fictional company, Beacon Desk combines two issue probabilities by adding them and labels the result “any issue.” Some requests have both issues, so the displayed estimate is inflated. A review separates the four disjoint categories and corrects the calculation.
The team then investigates whether its underlying estimates match recent observations. Fixing the arithmetic alone cannot answer that empirical question. This invented case distinguishes a mathematically coherent probability model from a model that has been validated for a real operational setting.

Common misconceptions and mistakes

Mutually exclusive events and independent events are different. Two disjoint events with positive probabilities cannot occur together, whereas independent events satisfy a product relationship for their joint probability. Do not interchange those conditions.
The axioms do not require equally likely outcomes or establish a particular interpretation of probability. They also do not justify adding probabilities of overlapping events without adjustment. In continuous models, a zero-probability event need not be logically impossible, so that phrase requires care. Countable additivity is specifically about countable collections; it is not permission to add probabilities over any arbitrary uncountable collection of singleton outcomes.

Business use

The axioms provide basic consistency checks for forecasts, scenarios, simulations, and uncertainty estimates. Teams can use them to catch negative values, impossible totals, double-counted categories, and incompatible event definitions before acting on a report.
Start by defining the sample space and clarifying whether categories overlap or cover every case. Keep the mathematical checks separate from questions about calibration, data quality, and changing conditions. A model can pass every arithmetic check while predicting poorly. Conversely, an apparently plausible forecast can contain internal contradictions that make its combined probabilities unreliable. Both levels of review are necessary for useful decision support.

Check your understanding

Why is the probability of either service in the worked example 0.45 rather than 0.55?

Answer: The 0.10 probability of needing both services appears in each individual event probability. Subtracting it once removes the double count.

Related terms

Sample space
Event
Mutually Exclusive Events
Independence
Joint probability
probability measure
countable additivity
complement

Learn next

Create a two-event table with four disjoint cells and assign probabilities totaling one. Recover each event probability and their union from the table. Then test independence separately by comparing the joint probability with the product of the marginal probabilities.

Sources

Sources: MIT OpenCourseWare, Introduction to Probability, Probability Models and Axioms, original course slides: https://ocw.mit.edu/courses/res-6-012-introduction-to-probability-spring-2018/4da14680987c076bd5aa2860da2d1afb_MITRES_6_012S18_L01AS.pdf

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