Russia 10 класс Probability and Statistics (advanced)

Chapters: 4

1. Graphs

Graphs

Graph Theory: Dots, Lines and Networks – A graph is a set of vertices (dots) joined by edges (lines). The degree of a vertex is how many edges touch it, and the sum of all degrees is twice the number of edges. An Euler trail uses every edge once and exists only when 0 or 2 vertices have odd degree. A tree is a connected graph with no cycles and n − 1 edges. Weighted graphs model roads and networks; Kruskal’s and Prim’s algorithms find a minimum spanning tree.

Events and operations · Conditional probability and Bayes

Probability: Events, Algebra of Events and Axioms – An event is any subset of the sample space S. From events A and B we build new events: not A (A′), A and B (A ∩ B), A or B (A ∪ B). Events are mutually exclusive if they share no outcome and exhaustive if together they cover S. The axiomatic approach says every P(E) ≥ 0, P(S) = 1, and for mutually exclusive A, B, P(A ∪ B) = P(A) + P(B). From these follow P(A′) = 1 − P(A) and P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Conditional Probability, Multiplication Rule and Independent Events – Conditional probability is the chance of A when we already know B has happened. We throw away every outcome outside B and count again: P(A|B) = P(A ∩ B) ÷ P(B). Turned around, this gives the multiplication rule P(A ∩ B) = P(B)·P(A|B). If knowing B does not change the chance of A, the events are independent and P(A ∩ B) = P(A)·P(B).

Bernoulli trials

Binomial Distribution – Repeat the same yes/no trial n times, independently, with the same chance of success p each time. The number of successes X follows the binomial distribution B(n, p): P(X = k) = ⁿCₖ pᵏ qⁿ⁻ᵏ with q = 1 − p. Its mean is np and its variance is npq.

Distributions · Expectation and variance

Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.