Probability
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
Probability foundations
- Sample spaces, events, and axioms of probability
- Combinatorial probability: permutations and combinations
- Conditional probability and independence
- Bayes' theorem and law of total probability
Random variables
- Random variables: discrete and continuous
- Probability mass functions and probability density functions
- Expected value, variance, and standard deviation
- Joint distributions; marginal and conditional distributions
- Covariance and correlation
Common distributions and limit theorems
- Common discrete distributions: Bernoulli, binomial, geometric, Poisson
- Common continuous distributions: uniform, exponential, normal
- Central Limit Theorem (statement) and normal approximations
- Sampling distributions (introduction for statistics follow-on)
STEM / applied
Engineering and statistics applications
- Reliability and quality control applications
- Queuing and Poisson process models (introduction)
- Engineering noise and signal: normal models
- Bio/statistics applications: hypothesis testing preview with probability models
Simulation and interpretation
- Monte Carlo simulation (introduction)
- Simulation with technology to estimate probabilities and expectations
- Risk analysis and expected utility (optional)
- Interpretation of p-values and confidence in context (bridge to statistics)
Notes
Topics reflect common undergraduate probability syllabi at US colleges and universities. Measure-theoretic probability is typically graduate-level; this list targets the standard calculus-based or algebra-based probability course.