Computer Science › Foundations of Computer Science
The set of every possible outcome of an experiment (the sample space), and an event as a subset of it — the vocabulary every probability statement is built from.
The three axioms (non-negativity, normalization, additivity) that any function must satisfy to count as a probability measure — and the rules (like unions and complements) that follow from them.
A function mapping outcomes to numbers, and the probability mass function (PMF) that describes exactly how likely each of its possible values is.
The long-run average value a random variable produces (expectation), and how spread out its values typically are around that average (variance).
Summarizing a real dataset you've actually collected — sample mean, median, and sample variance — and how these mirror, but are not identical to, the expectation and variance of a random variable.
Why the average of many independent samples converges to the true expectation as the sample count grows — the mathematical reason "averaging out noise" actually works.
A single yes/no trial (Bernoulli), and the count of successes across many independent repetitions of it (Binomial) — whose formula leans directly on the combinations already covered in discrete math.
The distribution of the number of independent events occurring in a fixed interval, derived as the limit of the Binomial distribution as the number of trials grows and success probability shrinks.
Describing two or more random variables together with a single joint PMF, and what it means for random variables — not just events — to be independent of one another.
A random variable that can take any value in a continuous range, described by a probability density function (PDF) whose height is not itself a probability — only the area under it is.
The bell-shaped distribution that shows up constantly in nature and statistics — and the reason it shows up so often is exactly the Central Limit Theorem covered next.
Why the sum (or average) of many independent random variables tends toward a Normal distribution, regardless of the shape of the individual variables being summed.
Why a statistic computed from a random sample is itself a random variable with its own distribution — and how the Central Limit Theorem predicts that distribution's shape.
Using a single number computed from data to estimate an unknown quantity about the population it came from, and what makes an estimator (like the sample mean) a good one.
A range of plausible values for an unknown quantity, built from a sample, that would contain the true value a specified percentage of the time if the sampling were repeated.
Formally testing a claim about data by asking how surprising the observed result would be if a specific "null" hypothesis were true — and what a p-value does and doesn't tell you.
A continuous distribution where every value in an interval is equally likely — the simplest possible PDF, and a natural first example of integrating a density to get a probability.
How the probability of an event updates once another event is known to have occurred, and the special case — independence — where knowing one event tells you nothing about the other.
Reversing a conditional probability — going from P(evidence | cause) to P(cause | evidence) — the single most consequential formula in this discipline for spam filters, diagnostics, and machine learning alike.
Two different philosophies for reasoning under uncertainty from data — updating a prior belief with Bayes' theorem versus treating the unknown quantity as fixed and the data as random — and when each is the more natural fit.
Part of the Computer Science curriculum.
The set of every possible outcome of an experiment (the sample space), and an event as a subset of it — the vocabulary every probability statement is built from.
The three axioms (non-negativity, normalization, additivity) that any function must satisfy to count as a probability measure — and the rules (like unions and complements) that follow from them.
A function mapping outcomes to numbers, and the probability mass function (PMF) that describes exactly how likely each of its possible values is.
The long-run average value a random variable produces (expectation), and how spread out its values typically are around that average (variance).
Summarizing a real dataset you've actually collected — sample mean, median, and sample variance — and how these mirror, but are not identical to, the expectation and variance of a random variable.
Why the average of many independent samples converges to the true expectation as the sample count grows — the mathematical reason "averaging out noise" actually works.
A single yes/no trial (Bernoulli), and the count of successes across many independent repetitions of it (Binomial) — whose formula leans directly on the combinations already covered in discrete math.
The distribution of the number of independent events occurring in a fixed interval, derived as the limit of the Binomial distribution as the number of trials grows and success probability shrinks.
Describing two or more random variables together with a single joint PMF, and what it means for random variables — not just events — to be independent of one another.
A random variable that can take any value in a continuous range, described by a probability density function (PDF) whose height is not itself a probability — only the area under it is.
The bell-shaped distribution that shows up constantly in nature and statistics — and the reason it shows up so often is exactly the Central Limit Theorem covered next.
Why the sum (or average) of many independent random variables tends toward a Normal distribution, regardless of the shape of the individual variables being summed.
Why a statistic computed from a random sample is itself a random variable with its own distribution — and how the Central Limit Theorem predicts that distribution's shape.
Using a single number computed from data to estimate an unknown quantity about the population it came from, and what makes an estimator (like the sample mean) a good one.
A range of plausible values for an unknown quantity, built from a sample, that would contain the true value a specified percentage of the time if the sampling were repeated.
Formally testing a claim about data by asking how surprising the observed result would be if a specific "null" hypothesis were true — and what a p-value does and doesn't tell you.
A continuous distribution where every value in an interval is equally likely — the simplest possible PDF, and a natural first example of integrating a density to get a probability.
How the probability of an event updates once another event is known to have occurred, and the special case — independence — where knowing one event tells you nothing about the other.
Reversing a conditional probability — going from P(evidence | cause) to P(cause | evidence) — the single most consequential formula in this discipline for spam filters, diagnostics, and machine learning alike.
Two different philosophies for reasoning under uncertainty from data — updating a prior belief with Bayes' theorem versus treating the unknown quantity as fixed and the data as random — and when each is the more natural fit.