1
Compute a probability from a sample space
Identify the outcomes, check they are equally likely, count the event, and divide — with the counting techniques of Combinatorics and Counting.
Course · Discrete Mathematics
Discrete Probability
Counting with a denominator. Every probability here is a ratio of counts, so Combinatorics and Counting does most of the work — and the parts that go wrong, conditioning and independence, go wrong for reasons that have nothing to do with arithmetic.
P(E) = |E| / |S| equally likely outcomes
P(A | B) = P(A ∩ B) / P(B)
independent ⟺ P(A ∩ B) = P(A)·P(B)
E[X + Y] = E[X] + E[Y] always, independent or not
What this course is about
- Subject
- Discrete Mathematics
- Lessons
- 12
- Format
- Interactive
Overview
What you will be able to do
By the end you can compute a probability exactly, condition correctly, say what an expectation does and does not tell you, and build a distribution from its sample space — or bypass it with indicators.
2
Condition without falling into the standard traps
P(A|B) and P(B|A) are different numbers, and the base rate is what separates them.
3
Use Bayes' theorem on a real question
The medical-test calculation, done properly, and the reason its answer surprises almost everyone.
4
Compute expectation and variance
From the definition, and by linearity — which holds whether or not the variables are independent, and is the most useful fact in the course.
5
Build a distribution, decompose a count, and tell the two distributions apart
Write P(X = k) from the sample space and check it sums to 1; split a count into indicators before reaching for a distribution; check the three binomial assumptions; and know that fixed trials is binomial and a fixed target is geometric.
Lessons
12 lessons
The model is set up first, then conditioning, and last random variables and the two distributions worth knowing by name.
Sample Spaces and EventsThe set of outcomes, and the subsets we ask about.Not marked
Computing ProbabilitiesCounting the event, counting the space, dividing.Not marked
The Probability AxiomsThree rules that every probability assignment obeys.Not marked
Conditional ProbabilityRestricting the sample space to what you now know.Not marked
IndependenceP(A ∩ B) = P(A)P(B), and nothing more.Not marked
Bayes' TheoremReversing a conditional, and why the base rate dominates.Not marked
Random VariablesA number attached to each outcome.Not marked
Expected ValueThe probability-weighted average, and what it does not mean.Not marked
Linearity of ExpectationE[X + Y] = E[X] + E[Y], whether or not they are independent.Not marked
Variance and Standard DeviationHow far from the mean, on average — squared.Not marked
Bernoulli Trials and the Binomial DistributionCounting successes in a fixed number of independent trials.Not marked
The Geometric Distribution and Waiting TimesHow long until the first success — and why the average misleads.Not marked
Recommended background
Counting, sets and functions.
Practice suggestions
Write the sample space down for the opening lessons, as far as “The Probability Axioms”, even when it is obvious. Most probability errors are errors about what the outcomes are.
Use the probability lab's outcome list. Every probability on this course is a count divided by a count, and seeing the outcomes highlighted makes conditioning concrete rather than formal. From “Random Variables” the distribution lab opens on each lesson's own worked example, so the numbers on the page are the numbers in the lab.
Do the Bayes calculation by frequencies as well as by the formula. The two agree, and the frequency version is the one that makes the answer believable. “Bayes' Theorem”’s lab does both on a million people, with every cell a whole number.
What it does not cover
Continuous probability: densities, the normal distribution, and anything requiring an integral. Everything here is finite or countable.
Limit theorems. The law of large numbers and the central limit theorem are mentioned where they explain something and are not proved.
Markov chains, martingales and stochastic processes, which are the natural sequel and a course of their own.