Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence, measured by the ratio of the favorable cases to the whole number of cases possible.

Discrete probability -the sample spaces have either finitely many or accountably many outcomes.

Continuos probabily= there is a continuum of possible outcomes.

Combinatorics a branch of discrete mathematics which is described as the art of arranging objects according to specified rules.

Combination - the number of different ways that a certain number of objects as a group can be selected from a larger number of objects.

Permutation is the number of different ways that a certain number of objects can be arranged in order from a larger number of objects.

4. Enumerative combinatorics

-Enumerative combinatorics is the most classical area of combinatorics, and concentrates on counting the number of certain combinatorial objects.

Ex. Fibonacci numbers i

Analytic combinatorics

-Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis andprobability theory.

Ex. complex asymptotics, singularity analysis, saddle-point asymptotics,

Partition theory

-Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials.

Ex. Ferrers diagrams

Graph theory

-Graphs are basic objects in combinatorics. The questions range from counting (e.g., the number of graphs on n vertices with k edges) to structural (e.g., which graphs contain Hamiltonian cycles) to algebraic questions.

Ex. Petersen graph.

Design theory

-Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties.

Ex. C-K design theory or concept-knowledge theory

Finite geometry

-Finite geometry is the study of geometric systems having only a finite number of points. Structures analogous to those found in continuous geometries (Euclidean plane, real projective space, etc.) but defined combinatorially are the main items studied.

Ex. Euclidean plane, real projective space

Order theory

-Order theory is the study of partially ordered sets, both finite and infinite.

Ex. lattices and Boolean algebras.

Matroid theory

-Matroid theory abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear dependence relation.

Ex. geometry, topology,combinatorial optimization, network theory and coding theory

Extremal combinatorics

-Extremal combinatorics studies extremal questions on set systems. The types of questions addressed in this case are about the largest possible graph which satisfies certain properties.

Ex. Sperner's theorem

Probabilistic combinatorics

- Probabilistic methods are also used to determine the existence of combinatorial objects with certain prescribed properties (for which explicit examples might be difficult to find), simply by observing that the probability of randomly selecting an object with those properties is greater than 0.

Ex. analysis of algorithms in computer science, as well as classical probability, additive and probabilistic number theory,

Algebraic combinatorics

-Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.

Ex. Young diagram of apartition (5,4,1).

Combinatorics on words

-Combinatorics on words deals with formal languages. It arose independently within several branches of mathematics, including number theory, group theory andprobability.

Ex. classical Chomsky–Schützenberger hierarchy of classes of formal grammars

Geometric combinatorics

-Geometric combinatorics is related to convex and discrete geometry, in particular polyhedral combinatorics. It asks, for example, how many faces of each dimension can a convex polytopehave.

Ex. Cauchy theorem on rigidity of convex polytopes.

Topological combinatorics

-Combinatorial analogs of concepts and methods in topology are used to study graph coloring,fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory.

Ex. Borsuk-Ulam theorem. It has also been used to study complexity problems in linear decision tree algorithms and the Aanderaa–Karp–Rosenberg conjecture.

Arithmetic combinatorics

-Arithmetic combinatorics arose out of the interplay between number theory, combinatorics,ergodic theory and harmonic analysis.

Ex. Szemerédi's theorem, Green-Tao theorem and extensions

Infinitary combinatorics

-Infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. It is a part ofset theory, an area of mathematical logic, but uses tools and ideas from both set theory and extremal combinatorics.

Ex. continuous graphs and trees, extensions of Ramsey's theorem, andMartin's axiom.

a random variable is a variable that takes on numerical values as a result of a random experiement or measurement,associates a numerical vaue with eac possible outcome.R.V must have numerical value.

A discrete random variables has a finite number of values or an infinites sequence of values and the differences between the outcome are meaningful.

die throw can only have 1,2,3,4,5,6, and each is meaningfully different.

discrete- all the possible values of a variable can me listed or counted.

x is the number on the sides of the die.

x is the number of students in your school.

a continuous random variable has a nearly infinite number of outcomes that cannot be easily counted and the differences between the outcomes are not meaningful.

- can take on an infinite number of possible values, corresponding to every value in an interval.

all possible values cannot be listed or counted.

heigt of adult filipino males.

time to faiure (in tousands of hours) for a type of light bulb.

weight can be any value between any two values that are infinite values, so we cannot list the values of x.

the probability distribution of a discrete variable is a table(graph or formula) that specifies the values of variabe and its corresponding probabilities.

x = 1,2,3,4,5

p(x)= .1,.2,.4,.2,.1

for any continuous probaility distribution:

f(x0 is greater that or equal to 0 for all x

the area under the entire curve is equal to one

normal distribution

exponential distribution

uniform- f(x) is constant over the range possible vaues of x