01
Sample space
What outcomes are possible?
Define the experiment, the set of outcomes, and the events whose probabilities you want to study.
Outcomes · distributions · conditioning · expectation
Probability Theory
Probability gives mathematical structure to uncertainty. It defines possible outcomes, assigns probabilities consistently, updates those probabilities when information changes, and studies distributions and long-run behavior without pretending individual random events become certain.
Probability workflow
Start with what could happen before asking how likely it is.
Probability models are built from a sample space and rules for assigning probability. Counting formulas, simulations, and familiar percentages are techniques inside that structure, not the definition of the subject.
02
Distribution
How is probability allocated?
A probability model assigns coherent weights to events or values. Equal likelihood is a special case, not a universal assumption.
03
Conditioning
What changes after information arrives?
Conditional probability restricts attention to outcomes consistent with known information and renormalizes uncertainty within that context.
04
Repeated behavior
What stabilizes over many trials?
Long-run frequencies and averages can approach model quantities even though individual outcomes remain uncertain.
Monte Carlo · area as probability
Random points can estimate π because geometry becomes an event probability.
The background samples points uniformly from a square whose inscribed circle has radius r. The area ratio is P(inside) = πr² / (2r)² = π/4, so the estimator is π̂ = 4 × inside / total. With independent identically distributed samples, the sample proportion tends toward the true event probability as the trial count grows.
Long-run convergence describes aggregate behavior. It does not make the next random point predictable.
A common special case
P(A) = |A| / |Ω|
This counting formula applies when Ω is finite and all elementary outcomes are equally likely. General probability theory does not require outcomes to be equally likely, countable, or even discrete.
Bayes’ rule
P(A|B) = P(B|A)P(A) / P(B)
For P(B) > 0, Bayes’ rule reverses conditioning by combining a likelihood with prior probability.
Monty Hall conditional-probability lab
Standard rules: the host knows the prize, always reveals an unchosen empty door, and always offers the switch.
Choose one door
Reference concepts
Uncertainty has structure, vocabulary, and algebra.
Precise definitions matter because everyday words such as “random,” “expected,” and “independent” have narrower mathematical meanings here.
01Event
A set of outcomes from the sample space. Probability is assigned to events, not only to single outcomes.
02Random variable
A function that maps outcomes to numerical values, letting probability models describe quantities such as counts, times, or measurements.
03Expected value
A probability-weighted average of possible values. It describes the center of a distribution, not a guarantee for any one trial.
04Independence
Two events are independent when learning that one occurred does not change the probability of the other: P(A|B) = P(A), when defined.