Expectation
A probability-weighted average describing long-run center.
Probability & Data · Accessible first encounter
Probability Theory:
How Information Changes Chance
Probability theory provides a mathematical language for uncertain events, random variables, expectation, dependence, and long-run behavior. It supports statistics, finance, science, algorithms, and decision-making.
01 · Opening mystery
Why does opening a losing door change the game?
In the Monty Hall problem, you choose one of three doors. The host, who knows where the prize is, opens a different door showing no prize. Two doors remain, so it feels as if each should have probability 1/2.
The host’s action is not random ignorance; it carries information. Your original choice still had probability 1/3, while the unchosen pair collectively had probability 2/3. The reveal concentrates that 2/3 on the remaining alternative.
Before exploringCan a simulation expose a mistaken intuition without replacing the proof?
Make a prediction. The laboratory is designed to challenge or refine it.
02 · Interactive laboratory
Compare staying and switching over many games.
Run fresh Monty Hall simulations. Small runs fluctuate; large runs settle near the theoretical probabilities 1/3 and 2/3.
Stay win rate33.3%
Switch win rate66.7%
03 · The big idea
Conditional probability changes the denominator.
The conditional probability P(A | B) asks for the probability of A within the reduced world where B has occurred. Its formula divides the probability of both A and B by the probability of B.
This framework distinguishes dependence from independence. Events A and B are independent when learning B does not change the probability of A.
Central definition
Conditional probability is the probability of A after restricting attention to outcomes in B.
P(A | B) = P(A ∩ B) / P(B), when P(B) > 0
Independence
Learning one event does not change the probability of the other.
Bayesian update
Combines prior belief with evidence to produce a posterior probability.
04 · A beautiful result
Bayes’ rule reverses the direction of conditioning.
Often we know how likely evidence is under a proposed cause, P(E | C), but want the probability of the cause after seeing the evidence, P(C | E). Bayes’ rule connects the two.
The prior P(C) matters. A highly accurate medical test can still produce many false alarms when the condition is very rare because most tested people begin in the no-condition group.
- 1
By the conditional formula, P(C | E) = P(C ∩ E)/P(E).
- 2
Also P(E | C) = P(C ∩ E)/P(C).
- 3
Therefore P(C ∩ E) = P(E | C)P(C).
- 4
Substitute to obtain P(C | E) = P(E | C)P(C)/P(E).
05 · Why this subject matters
Probability separates randomness from ignorance.
Probability models repeated experiments, measurement noise, reliability, queues, genetics, insurance claims, market risk, and randomized algorithms. A mathematical model states what is random and how outcomes are weighted.
Advanced probability studies random variables, distributions, convergence, stochastic processes, martingales, and measure-theoretic foundations.
Inference
Learns about populations from uncertain samples.
Risk Models
Describes uncertain returns and contingent payoffs.
Stochastic Processes
Models systems evolving through random transitions.
06 · Friendly assessment
Check the central ideas without pressure.
The questions focus on the main insights, not obscure details. Each response receives an explanation immediately.
Where this idea leads
Continue through the mathematical atlas.
Connected field
Mathematical Statistics
Turn probability models into inference from data.
Explore →Connected fieldStochastic Processes
Study random behavior through time.
Explore →Connected fieldActuarial Mathematics
Pool and price uncertain future claims.
Explore →Connected fieldQuantitative Finance
Value and manage uncertain financial payoffs.
Explore →You have now experienced
You have used conditional probability to resolve a paradox and seen long-run simulation approach an exact theoretical value.
This is an invitation to continue, not a compressed substitute for a full university course.