List or characterize possible outcomes.
The sample space can be finite, countable or continuous depending on the problem.
Side 134
Probability Theory
The mathematics of uncertainty: formal structures for representing possible outcomes, dependence, random quantities and what happens when random experiments repeat.
sample space→random variable→conditioning→expectation→limits
04lenses
16working concepts
V0content
SS-1.0standard
Probability starts with a space of possible outcomes.
Events are sets, probabilities are measures, and the model only becomes meaningful after the experiment and its assumptions are defined.
Represent a proposition as a set of outcomes.
Set operations translate directly into combinations of probabilistic statements.
Assign coherent weights to events.
Probabilities obey normalization and additivity rather than arbitrary scoring.
Separate events whose joint probability factors.
Independence is a structural assumption and should not be confused with zero correlation.
Random variables turn outcomes into quantities.
Distributions describe how probability is carried through a numerical transformation of the sample space.
Map outcomes to numbers.
The map makes arithmetic and expectation possible while preserving uncertainty.
Describe probability over values.
Mass functions, densities and cumulative distributions encode the same random quantity in different forms.
Average with respect to probability.
Expected value is a weighted mean, not the value one should expect to observe in a single trial.
Measure squared spread around the mean.
Variance summarizes dispersion but does not fully describe distribution shape.
New information changes the relevant probability space.
Conditional probability is the formal basis of learning from partial information.
Restrict attention to outcomes compatible with evidence.
Conditioning renormalizes probabilities inside the observed event.
Reverse a conditional relationship.
Bayes' rule is an identity; substantive inference still depends on the chosen model.
Average given available information.
This object supports prediction, martingales and sequential decision-making.
Model how variables move together.
Joint and conditional distributions contain more information than pairwise correlation alone.
Repeated randomness can produce stable large-scale regularities.
Limit theorems explain why frequencies, averages and standardized sums often become predictable.
Sample averages stabilize under conditions.
Repeated observations can converge toward an expected value without individual outcomes becoming less random.
Many normalized sums approach Gaussian form.
The theorem explains recurring normal approximations while depending on specific conditions.
Distinguish modes of convergence.
Probability, distribution and almost-sure convergence answer different asymptotic questions.
Rare outcomes can dominate risk.
Heavy-tailed distributions make extreme events more important than Gaussian intuition suggests.
Probability is structure before it is prediction. A rigorous model begins by defining what can happen, how events are measured and what assumptions connect one random quantity to another.