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Sample Space and Outcomes
Every probability model starts with a sample space — a description of all possible things that may happen during an experiment.
Intro
A vector is an arrow with direction and magnitude.
Events
An event is a subset of the sample space — a collection of outcomes that we assign probability to. We assign probabilities to events, not individual outcomes. This matters especially for continuous sample spaces (where individual outcomes have probability zero) and compound outcomes like "the sum of two dice is 7".
Conditional Probability
Conditional probability measures the likelihood of an event given that another event has occurred.
Continuous Probability
For continuous sample spaces (like a dart hitting a target), we can't list or count outcomes. Instead, probability is defined by area (or length, or volume).
Total Probability Theorem
Suppose you want the probability that a widget is defective. You could try to count defective widgets directly — but what if the factory has multiple machines, each with a different defect rate, and you don't know which machine made a given widget? You can't just use one defect rate; you need to account for all of them.
Row Picture
The row picture visualizes each equation as a line.
Probability Axioms
Every event has a probability. For example, P(A) means "the probability of event A happening." A rule that assigns such probabilities to all events is called a probability law. All probability laws must satisfy three fundamental axioms, formalized by Andrey Kolmogorov in 1933.
Discrete Uniform Law
When all outcomes in a finite sample space are equally likely, probability reduces to counting.
Probability Trees
Probability trees visualize multi-stage experiments with conditional probabilities.
Scalar
Multiply a vector by a scalar (number):
Addition
To add two vectors, add their components:
Key Terms
93 termsA state i where P(i→i) = 1 — once entered, the chain never leaves.
The probability that the chain eventually enters a particular absorbing state.
A Markov chain where the GCD of return times to any state is 1 (no cycling behavior).
$P(A) = \text{area of } A \,/\, \text{area of } \Omega$
A fundamental rule accepted without proof
$\pi_j = \sum_i \pi_i \cdot P(i \to j)$ — the equations that define the steady-state distribution.
$P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$ — updates the probability of B after observing A.
A random variable with exactly two outcomes: success (1) with probability p, and failure (0) with probability 1−p.
The number of successes in n independent Bernoulli trials, each with success probability p.
Cumulative Distribution Function — $F(x) = P(X \leq x)$, always non-decreasing with $F(-\infty)=0$ and $F(\infty)=1$.
The sum (or mean) of many independent random variables is approximately normal, regardless of their individual distributions.
Extends the multiplication rule: $P(A \cap B \cap C) = P(A) \times P(B|A) \times P(C|A \cap B)$, and so on for more events.