PELUANG

Peluang atau kebolehjadian atau dikenal juga sebagai probabilitas adalah cara untuk mengungkapkan pengetahuan atau kepercayaan bahwa suatu kejadian akan berlaku atau telah terjadi. Konsep ini telah dirumuskan dengan lebih ketat dalam matematika, dan kemudian digunakan secara lebih luas dalam tidak hanya dalam matematika atau statistika, tapi juga keuangan, sains dan filsafat.

Konsep matematika

Probabilitas suatu kejadian adalah angka yang menunjukkan kemungkinan terjadinya suatu kejadian. Nilainya di antara 0 dan 1. Kejadian yang mempunyai nilai probabilitas 1 adalah kejadian yang pasti terjadi atau sesuatu yang telah terjadi^[1]. Misalnya matahari yang masih terbit di timur sampai sekarang. Sedangkan suatu kejadian yang mempunyai nilai probabilitas 0 adalah kejadian yang mustahil atau tidak mungkin terjadi. Misalnya seekor kambing melahirkan seekor sapi.
Probabilitas/Peluang suatu kejadian A terjadi dilambangkan dengan notasi P(A), p(A), atau Pr(A). Sebaliknya, probabilitas [bukan A] atau komplemen A, atau probabilitas suatu kejadian A tidak akan terjadi, adalah 1-P(A). Sebagai contoh, peluang untuk tidak munculnya mata dadu enam bila sebuah dadu
bersisi enam digulirkan adalah

Peluang, Permutasi & Kombinasi Matematika

Rumus Web mengumpulkan materi Peluang, Permutasi & Kombinasi Matematika ini untuk anak SMA demi UAN SNMPTN SPMB SIMAK UI. Silakan dipelajari

1) Permutasi
Permutasi adalah susunan unsur-unsur yang berbeda dalam urutan tertentu. Pada permutasi urutan diperhatikan sehingga
Permutasi k unsur dari n unsur adalah semua urutan yang berbeda yang mungkin dari k unsur yang diambil dari n unsur yang berbeda. Banyak permutasi k unsur dari n unsur ditulis atau .
Permutasi siklis (melingkar) dari n unsur adalah (n-1) !
Cara cepat mengerjakan soal permutasi

dengan penulisan nPk, hitung 10P4
kita langsung tulis 4 angka dari 10 mundur, yaitu 10.9.8.7
jadi 10P4 = 10x9x8x7 berapa itu? hitung sendiri

Contoh permutasi siklis :

Suatu keluarga yang terdiri atas 6 orang duduk mengelilingi sebuah meja makan yang berbentuk lingkaran. Berapa banyak cara agar mereka dapat duduk mengelilingi meja makan dengan cara yang berbeda?
Jawab :
Banyaknya cara agar 6 orang dapat duduk mengelilingi meja makan dengan urutan yang berbeda sama dengan banyak permutasi siklis (melingkar) 6 unsur yaitu :

2) Kombinasi
Kombinasi adalah susunan unsur-unsur dengan tidak memperhatikan urutannya. Pada kombinasi AB = BA. Dari suatu himpunan dengan n unsur dapat disusun himpunan bagiannya dengan untuk Setiap himpunan bagian dengan k unsur dari himpunan dengan unsur n disebut kombinasi k unsur dari n yang dilambangkan dengan ,
Contoh :
Diketahui himpunan .
Tentukan banyak himpunan bagian dari himpunan A yang memiliki 2 unsur!
Jawab :

Banyak himpunan bagian dari A yang memiliki 2 unsur adalah C (6, 2).

Cara cepat mengerjakan soal kombinasi

dengan penulisan nCk, hitung 10C4

kita langsung tulis 4 angka dari 10 mundur lalu dibagi 4!, yaitu 10.9.8.7 dibagi 4.3.2.1
jadi 10C4 = 10x9x8x7 / 4x3x2x1 berapa itu? hitung sendiri
Ohya jika ditanya 10C6 maka sama dengan 10C4, ingat 10C6=10C4. contoh lainnya
20C5=20C15
3C2=3C1
100C97=100C3
melihat polanya? hehe semoga bermanfaat!

Peluang Matematika

1. Pengertian Ruang Sampel dan Kejadian
Himpunan S dari semua kejadian atau peristiwa yang mungkin mucul dari suatu percobaan disebut ruang sampel. Kejadian khusus atau suatu unsur dari S disebut titik sampel atau sampel. Suatu kejadian A adalah suatu himpunan bagian dari ruang sampel S.

Contoh:
Diberikan percobaan pelemparan 3 mata uang logam sekaligus 1 kali, yang masing-masing memiliki sisi angka ( A ) dan gambar ( G ). Jika P adalah kejadian muncul dua angka, tentukan S, P (kejadian)!
Jawab :
S = { AAA, AAG, AGA, GAA, GAG, AGG, GGA, GGG}
P = {AAG, AGA, GAA}

2. Pengertian Peluang Suatu Kejadian
Pada suatu percobaan terdapat n hasil yang mungkin dan masing-masing berkesempatan sama untuk muncul. Jika dari hasil percobaan ini terdapat k hasil yang merupakan kejadian A, maka peluang kejadian A ditulis P ( A ) ditentukan dengan rumus :

Contoh :
Pada percobaan pelemparan sebuah dadu, tentukanlah peluang percobaan kejadian muncul bilangan genap!
Jawab : S = { 1, 2, 3, 4, 5, 6} maka n ( S ) = 6
Misalkan A adalah kejadian muncul bilangan genap, maka:
A = {2, 4, 6} dan n ( A ) = 3

3. Kisaran Nilai Peluang Matematika
Misalkan A adalah sebarang kejadian pada ruang sampel S dengan n ( S ) = n, n ( A ) = k dan
Jadi, peluang suatu kejadian terletak pada interval tertutup [0,1]. Suatu kejadian yang peluangnya nol dinamakan kejadian mustahil dan kejadian yang peluangnya 1 dinamakan kejadian pasti.
4. Frekuensi Harapan Suatu Kejadian
Jika A adalah suatu kejadian pada frekuensi ruang sampel S dengan peluang P ( A ), maka frekuensi harapan kejadian A dari n kali percobaan adalah n x P( A ).

Contoh :
Bila sebuah dadu dilempar 720 kali, berapakah frekuensi harapan dari munculnya mata dadu 1? Jawab :
Pada pelemparan dadu 1 kali, S = { 1, 2, 3, 4, 5, 6 } maka n (S) = 6.
Misalkan A adalah kejadian munculnya mata dadu 1, maka:
A = { 1 } dan n ( A ) sehingga :

Frekuensi harapan munculnya mata dadu 1 adalah

5. Peluang Komplemen Suatu Kejadian
Misalkan S adalah ruang sampel dengan n ( S ) = n, A adalah kejadian pada ruang sampel S, dengan n ( A ) = k dan Ac adalah komplemen kejadian A, maka nilai n (Ac) = n – k, sehingga :

Jadi, jika peluang hasil dari suatu percobaan adalah P, maka peluang hasil itu tidak terjadi adalah (1 – P).

Peluang Kejadian Majemuk

1. Gabungan Dua Kejadian
Untuk setiap kejadian A dan B berlaku :
Catatan : dibaca “ Kejadian A atau B dan dibaca “Kejadian A dan B”

Contoh :
Pada pelemparan sebuah dadu, A adalah kejadian munculnya bilangan komposit dan B adalah kejadian muncul bilangan genap. Carilah peluang kejadian A atau B!
Jawab :

2. Kejadian-kejadian Saling Lepas
Untuk setiap kejadian berlaku Jika . Sehingga Dalam kasus ini, A dan B disebut dua kejadian saling lepas.

3. Kejadian Bersyarat
Jika P (B) adalah peluang kejadian B, maka P (A|B) didefinisikan sebagai peluang kejadian A dengan syarat B telah terjadi. Jika adalah peluang terjadinya A dan B, maka Dalam kasus ini, dua kejadian tersebut tidak saling bebas.

4. Teorema Bayes
Teorema Bayes(1720 – 1763) mengemukakan hubungan antara P (A|B) dengan P ( B|A ) dalam teorema berikut ini :
 
5. Kejadian saling bebas Stokhastik
(i) Misalkan A dan B adalah kejadian – kejadian pada ruang sampel S, A dan B disebut dua kejadian saling bebas stokhastik apabila kemunculan salah satu tidak dipengaruhi kemunculan yang lainnya atau : P (A | B) = P (A), sehingga:

Sebaran Peluang

1. Pengertian Peubah acak dan Sebaran Peluang.
Peubah acak X adalah fungsi dari suatu sampel S ke bilangan real R. Jika X adalah peubah acak pada ruang sampel S denga X (S) merupakan himpunan berhingga, peubah acak X dinamakan peubah acak diskrit. Jika Y adalah peubah acak pada ruang sampel S dengan Y(S) merupakan interval, peubah acak Y disebut peubah acak kontinu. Jika X adalah fungsi dari sampel S ke himpunan bilangan real R, untuk setiap dan setiap maka:

Misalkan X adalah peubah acak diskrit pada ruang sampel S, fungsi masa peluang disingkat sebaran peluang dari X adalah fungsi f dari R yang ditentukan dengan rumus berikut :



2. Sebaran Binom
Sebaran Binom atau Distribusi Binomial dinyatakan dengan rumus sebagai berikut :

Dengan P sebagai parameter dan
Rumus ini dinyatakan sebagai:
untuk n = 0, 1, 2, …. ,n

Dengan P sebagai parameter dan
P = Peluang sukses
n = Banyak percobaan
x = Muncul sukses
n-x = Muncul gagal

MATRIKS TRANSFORMASI

Definition and first consequences

Let V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

	additivity
	homogeneity of degree 1

This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors x₁, ..., x_m ∈ V and scalars a₁, ..., a_m ∈ K, the following equality holds:

Denoting the zeros of the vector spaces by 0, it follows that f(0) = 0 because letting α = 0 in the equation for homogeneity of degree 1,

f(0) = f(0 ⋅ 0) = 0 f(0) = 0.

Occasionally, V and W can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear.
A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional.
These statements generalize to any left-module _RM over a ring R without modification.

Examples

• The identity map and zero map are linear.
• The map , where c is a constant, is linear.
• For real numbers, the map is not linear.
• For real numbers, the map is not linear (but is an affine transformation, and also a linear function, as defined in analytic geometry.)
• If A is a real m × n matrix, then A defines a linear map from Rⁿ to R^m by sending the column vector x ∈ Rⁿ to the column vector Ax ∈ R^m. Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section.
• The (definite) integral is a linear map from the space of all real-valued integrable functions on some interval to R
• The (indefinite) integral (or antiderivative) is not considered a linear transformation, as the use of a constant of integration results in an infinite number of outputs per input.
• Differentiation is a linear map from the space of all differentiable functions to the space of all functions.
• If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dim_F(W) × dim_F(V) matrices in the way described in the sequel are themselves linear maps.
• The expected value of a random variable is linear, as for random variables X and Y we have E[X + Y] = E[X] + E[Y] and E[aX] = aE[X], but the variance of a random variable is not linear, as it violates the second condition, homogeneity of degree 1: V[aX] = a²V[X].

Matrices

Main article: Transformation matrix

If V and W are finite-dimensional, and one has chosen bases in those spaces, then every linear map from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear maps: if A is a real m × n matrix, then the rule f(x) = Ax describes a linear map Rⁿ → R^m (see Euclidean space).
Let {v₁, ..., v_n} be a basis for V. Then every vector v in V is uniquely determined by the coefficients c₁, ..., c_n in

If f: V → W is a linear map,

which implies that the function f is entirely determined by the values of f(v₁), ..., f(v_n).
Now let {w₁, ..., w_m} be a basis for W. Then we can represent the values of each f(v_j) as

Thus, the function f is entirely determined by the values of a_ij.
If we put these values into an n × m matrix M, then we can conveniently use it to compute the value of f for any vector in V. For if we place the values of c₁, ..., c_n in an n × 1 matrix C, we have MC = the m × 1 matrix whose ith element is the coordinate of f(v) which belongs to the base w_i.
A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.

Examples of linear transformation matrices

In two-dimensional space R² linear maps are described by 2 × 2 real matrices. These are some examples:

• rotation by 90 degrees counterclockwise:

  

• rotation by θ degrees counterclockwise:

  

• reflection against the x axis:

  

• reflection against the y axis:

  

• scaling by 2 in all directions:

  

• horizontal shear mapping:

  

• squeeze mapping:

  

• projection onto the y axis:

  

Forming new linear maps from given ones

The composition of linear maps is linear: if f: V → W and g: W → Z are linear, then so is their composition g o f: V → Z. It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category.
The inverse of a linear map, when defined, is again a linear map.
If f₁: V → W and f₂: V → W are linear, then so is their sum f₁ + f₂ (which is defined by (f₁ + f₂)(x) = f₁(x) + f₂(x)).
If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a (f(x)), is also linear.
Thus the set L(V, W) of linear maps from V to W itself forms a vector space over K, sometimes denoted Hom(V, W). Furthermore, in the case that V = W, this vector space (denoted End(V)) is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.
Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

Endomorphisms and automorphisms

A linear transformation f: V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity map id: V → V.
An endomorphism of V that is also an isomorphism is called an automorphism of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V).
If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.

Kernel, image and the rank–nullity theorem

If f : V → W is linear, we define the kernel and the image or range of f by

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is known as the rank–nullity theorem:

The number dim(im(f)) is also called the rank of f and written as rank(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as null(f) or ν(f). If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.

Cokernel

Main article: Cokernel

A subtler invariant of a linear transformation is the cokernel, which is defined as

This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space of the target. Formally, one has the exact sequence

These can be interpreted thus: given a linear equation f(v) = w to solve,

• the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of degrees of freedom in a solution, if it exists;
• the co-kernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space W/f(V) is the dimension of the target space minus the dimension of the image.
As a simple example, consider the map f: R² → R², given by f(x, y) = (0, y). Then for an equation f(x, y) = (a, b) to have a solution, we must have a = 0 (one constraint), and in that case the solution space is (x, b) or equivalently stated, (0, b) + (x, 0), (one degree of freedom). The kernel may be expressed as the subspace (x, 0) < V: the value of x is the freedom in a solution – while the cokernel may be expressed via the map W → R, given a vector (a, b) , the value of a is the obstruction to there being a solution.
An example illustrating the infinite-dimensional case is afforded by the map f: R^∞ → R^∞, with b₁ = 0 and b_{n + 1} = a_n for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ( ), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map h: R^∞ → R^∞, with c_n = a_{n + 1}. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.

Index

For a linear operator with finite-dimensional kernel and co-kernel, one may define index as:

namely the degrees of freedom minus the number of constraints.
For a transformation between finite-dimensional vector spaces, this is just the difference dim(V) − dim(W), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.
The index comes of its own in infinite dimensions: it is how homology is defined, which is a central theory in algebra and algebraic topology; the index of an operator is precisely the Euler characteristic of the 2-term complex 0 → V → W → 0. In operator theory, the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem.

Algebraic classifications of linear transformations

No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.
Let V and W denote vector spaces over a field, F. Let T: V → W be a linear map.

• T is said to be injective or a monomorphism if any of the following equivalent conditions are true:
  • T is one-to-one as a map of sets.
  • kerT = {0_V}
  • T is monic or left-cancellable, which is to say, for any vector space U and any pair of linear maps R: U → V and S: U → V, the equation TR = TS implies R = S.
  • T is left-invertible, which is to say there exists a linear map S: W → V such that ST is the identity map on V.

• T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:
  • T is onto as a map of sets.
  • coker T = {0_W}
  • T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: W → U and S: W → U, the equation RT = ST implies R = S.
  • T is right-invertible, which is to say there exists a linear map S: W → V such that TS is the identity map on W.

• T is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism.

• If T: V → V is an endomorphism, then:
  • If, for some positive integer n, the n-th iterate of T, Tⁿ, is identically zero, then T is said to be nilpotent.
  • If T² = T, then T is said to be idempotent
  • If T = kI, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map; see scalar matrix.

Change of basis

Given a linear map whose matrix is A, in the basis B of the space it transforms vectors coordinates [u] as [v] = A[u]. As vectors change with the inverse of B, its inverse transformation is [v] = B[v'].
Substituting this in the first expression

hence

Therefore the matrix in the new basis is A′ = B⁻¹AB, being B the matrix of the given basis.
Therefore linear maps are said to be 1-co 1-contra -variant objects, or type (1, 1) tensors.

Continuity

Main article: Discontinuous linear map

A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. An infinite-dimensional domain may have discontinuous linear operators.
An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, sin(nx)/n converges to 0, but its derivative cos(nx) does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

Applications

A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix.
Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.