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P1-6.2 Distinguishing Probability, Uncertainty, and Stochastic Processes

Section ID: P1-6.2 Version: v2026.07.20

Section 6.1 showed why some problems are difficult to handle with explicit rules alone. When information is incomplete, observations are unstable, and several outcomes remain possible, AI has to deal with uncertainty.

This section separates three expressions that are often mixed:

uncertainty = the state of not knowing for sure
probability = the numerical language used to express that uncertainty
stochastic = a process or property that includes probabilistic variation in the process itself

These are connected, but they are not the same thing.

In Part 1, this section fixes the baseline distinction among uncertainty, probability, and stochastic process. Section 6.1 first fixed the problem conditions that require those terms. Here we set the vocabulary needed to read those conditions. Search and heuristics return in P1-7, and fuller probability formulas return in Part 2 and Part 4.

Why This Distinction Is Needed

The sentence AI answers probabilistically looks simple on the surface, but it actually splits into several different questions:

Question Connected concept
would the judgment change if more information were observed? uncertainty, evidence
how much should we trust the number 0.70 produced by the model? probability, calibration
why do results vary slightly even when the same action is repeated? stochastic process
is missing knowledge the same thing as the world itself being variable? epistemic uncertainty, aleatoric uncertainty

Hüllermeier and Waegeman explain that uncertainty is a central methodological issue in machine learning, and that the distinction between aleatoric and epistemic uncertainty becomes increasingly important as practical deployment and safety demands grow. We do not go deeply into those two categories here. It is enough to keep the sense that uncertainty itself also has kinds.

The three baseline questions are:

is this a kind of not-knowing that decreases if we observe more?
is this a kind of variation that remains even when we repeat?
do the model’s numbers match real-world frequencies well?

Section 6.2 first closes how uncertainty, probability, and stochastic process differ. Conditional probability, Bayes’ rule, and probability-distribution calculations return in the statistics basics of Part 2 and the evaluation and prediction chapters of Part 4. LLM sampling and next-token prediction continue in P6-5.1 and P6-5.2 of Part 6. Here we set only the vocabulary baseline needed to read AI documents.

Separating Uncertainty, Probability, and Stochasticity

  • Understand uncertainty as a state of not knowing.
  • Understand probability as the numerical language used to express and update uncertainty.
  • Understand stochastic as a property of a process that includes probabilistic variation.
  • Avoid treating random, nondeterministic, and stochastic as if they were all the same.
  • Set a translation baseline so that uncertainty, probability, and related Korean expressions do not collapse into one.

Three Standards

Standard Why it matters Level of understanding needed here
uncertainty is the state of not yet knowing This prevents it from being mixed directly with numerical probability. Understand it as ignorance before a number is assigned.
probability is the numerical language that expresses that ignorance This is the starting point for reading model scores and uncertainty numerically. See it as a tool for comparing which candidate is more plausible.
stochastic means the process itself contains variation This reduces the mistake of using uncertainty, probability, and randomness as one word. Understand that not only results, but the process itself may vary.

A short role separation is useful:

Term Very short meaning Role in this section
uncertainty not knowing the state before a numerical expression
probability number the numerical expression of that state
stochastic process varying process a process whose behavior includes probabilistic variation
random arbitrary selection or randomness a word that must not be used too loosely
nondeterministic result not fixed to one outcome not always the same as probabilistic

Uncertainty Is the State of Not Knowing

Uncertainty means a state where some fact, state, or outcome cannot be stated with certainty.

Situation What is uncertain?
customer-support message whether the real intent is delivery or refund
sensor input whether the measured value matches the real distance
medical diagnosis whether symptoms alone identify the cause
delivery forecast whether it will arrive tomorrow or be delayed

Poole and Mackworth explain that agents in the real world do not know everything and must update beliefs from observations. The Stanford Encyclopedia of Philosophy entry on AI also notes that purely logic-based approaches cannot always determine truth or falsity for every proposition, because ignorance, physical indeterminacy, and ambiguity often require probabilistic handling.

uncertainty = a state that cannot yet be stated with confidence

Uncertainty is not itself a number. It is the problem state before the numerical expression.

Probability Is the Numerical Language of Uncertainty

Probability is the mathematical language used to handle uncertain states. Poole and Mackworth explain probability as a calculus of belief that can be updated when new information arrives.

Suppose a support-message classifier produces the following scores:

Candidate Score
delivery 0.70
refund 0.20
other 0.10

That number does not mean delivery is absolutely true. It is closer to saying that, given the model’s information and criteria, delivery currently looks more plausible.

Poole and Mackworth describe probability as a number between 0 and 1, where 0 corresponds to believing something false and 1 to believing it true. A value in between does not mean the fact is partially true. It means the agent does not know for sure whether it is true or false.

uncertainty: we do not know whether this is delivery or refund
probability: we express it as delivery 0.70, refund 0.20, other 0.10

So probability is not uncertainty itself. It is a way to express, compare, and compute with uncertainty.

How Much Can We Trust a Number Like 0.70?

If a model assigns 0.70, that does not automatically mean the number always matches a 70% real-world frequency. scikit-learn explains that in classification, people often want not only class labels but also probabilities, yet some models estimate class probabilities poorly.

That raises the question of calibration:

if we collect the cases where the model said 0.70,
are roughly 70% of those cases actually correct?

A well-calibrated classifier makes it more reasonable to read predicted probabilities as confidence-like numbers. For example, if a classifier repeatedly outputs values around 0.80 for many binary-classification samples, we can check whether the positive class actually occurs in roughly 80% of those cases.

So in this section, a score like delivery 0.70 should be read cautiously:

Hasty reading Safer reading
delivery is 70% true under the current data and model criteria, delivery was scored at 0.70
the model said 70%, so we can trust it we should separately check whether that model’s probability output is calibrated
0.70 is an absolute trust level its meaning depends on data, model, and evaluation method

Evidence Is Observed Information That Updates Belief

Poole and Mackworth describe observed information as evidence, and explain the perspective of updating prior probability into posterior probability based on that evidence.

In the support-message example:

Stage Example
prior judgment from tracking does not work alone, delivery seems likely
additional evidence payment logs show duplicate payment
updated judgment this may not be a simple delivery issue; payment must also be considered

We do not compute Bayes’ rule here. The important point is simply that as new observation arrives, the plausibility of different conclusions can change.

when observation changes,
the plausibility of different conclusions can also change

Stochastic Means the Process Includes Probabilistic Variation

Stochastic is often translated simply as probabilistic, but it is not identical to probability.

English expression Meaning used here Central idea
probability probability a number or distribution expressing uncertainty
probabilistic probabilistic using probability in expression or computation
stochastic stochastic process or stochastic property a process, action, or outcome includes probabilistic variation

Poole and Mackworth explain that actions can behave as stochastic processes. For example, a robot may overshoot a target slightly when moving, and teaching some material to a student does not guarantee the student will learn it every time.

The useful intuition is:

even if we repeat the same action,
the same result is not guaranteed every time

Examples:

Situation Why it can be read as stochastic
robot movement the same command can lead to slightly different positions because of friction, sensors, or floor condition
user response showing the same recommendation does not guarantee the same click behavior
LLM response generation the same prompt can produce different wording depending on generation settings and sampling

But stochastic does not mean completely arbitrary. It means the variation follows some probabilistic rule or distribution.

Random, Stochastic, and Nondeterministic Should Be Used Carefully

In Korean, words corresponding to random, stochastic, and nondeterministic are often mixed. In AI writing, it is safer to separate them.

English expression Preferred meaning here Caution
random arbitrary or random choice do not use it as if it meant pure chaos
stochastic a process that includes probabilistic variation a process may vary without being structureless
nondeterministic not fixed to one result for the same state and input it may not explicitly specify a probability distribution
probabilistic expressed or computed using probability may be the modeling language for a stochastic process

For example, if an LLM can answer differently to the same prompt, it is imprecise to say it answers any which way. The safer wording is:

depending on generation settings and sampling, the output may vary

A Note on Korean Wording

In Korean writing, words corresponding to uncertainty and indeterminacy sometimes appear together. In this book, for general AI explanation, uncertainty is preferred.

The reason is practical: uncertainty works well for missing information, incomplete observation, and predictive uncertainty. The other word can be read with stronger associations to specific fields such as physics. So the baseline here is:

Korean-side category English expression Use standard in this book
uncertainty uncertainty use in general AI contexts for states of not knowing
probability probability use for the numeric expression of that state
probabilistic or stochastic probabilistic, stochastic use when explaining probability-based modeling or varying processes

The goal is not to force language mechanically, but to help readers avoid mixing several levels together.

The Same Example Again

Take the support sentence again:

I ordered yesterday, but tracking still does not work.

Perspective Explanation
uncertainty we do not know whether the real cause is delivery delay, system update delay, or a payment issue
probability with current information, we may express it as delivery 0.70, payment 0.20, other 0.10
evidence extra observations such as order time, payment log, or carrier integration state
stochastic process delivery time or user response may vary probabilistically even under similar conditions

Keeping this distinction helps us read the sentence AI behaves probabilistically more carefully. Some part concerns uncertainty, some part concerns numerical probability, and some part concerns the stochastic nature of a process.

Checklist

  • You can explain uncertainty as a state of not knowing rather than as a number.
  • You can explain probability as the numerical language used to express and update uncertainty.
  • You can explain that evidence is observed information that updates a judgment.
  • You can explain stochastic as the property that a process or action includes probabilistic variation.
  • You can explain why random, stochastic, nondeterministic, and probabilistic should not be treated as the same word.
  • You can explain why general AI discussion should prefer uncertainty over collapsing several different ideas into one vague label.
  • You can distinguish whether a sentence is talking about a state of not knowing, a numerical expression, or a property of a process.
  • You can distinguish not-knowing that may shrink with more observation from variability that remains in the process itself.

Sources and Further Reading