Combinatorial analysis and probability
A TREE DIAGRAM A tree diagram is a graph that is born from the trunk and it’s branches. And it continues to branch out like a tree. A branch is every one of the arrows from the diagram. A way is a set of branches that go from the beginning to the end EXAMPLE: A general doctor classifies his patients in accordance with his gender (male M or female F), blood type (A,B,AB u O) and as for the blood pressure (normal N, high H or low L). By means of a tree diagram, how many classifications of patients can there be ?
Solución: • • • • • • • • • • • •
• • • •
M
F
A
N H L
B
N H L
AB
N H L
O
N H L
A
N H L
B
N H L
AB O
N H L N H L
ORDINARY VARIATIONS OR WITHOUT REPETITION The ordinary variations or without repetition from m elements taken from p and p, being p ≤ m, they are different groups from p distinct elements between themselves that can form with the m elements, so that, in every two groups: • the order of the elements is distinct or • Some of the elements are distinct They are represented by Vm,p or V , and there is had that
EXAMPLE : With the letters a, b and c form all of the words that you can without repeting none. Eventhough it doesnt make sense. V ₃,₂=3• 2=6 b ab a c ac
b
a c
ba bc
c
a b
ca cb
VARIATIONS WITH REPETITION. The variations with repetition of m elements taken from p and p, they are different groups that can be formed with the m elements, so that in every two groups: • the order o f the elements are distinct or • some of the elements are distinct They are represented by VRm,p ,and there is had that: VRm,p=m^p
EXAMPLE: With the letters a , b and c form all of the words that you can from the two eventhough it doesnt make sense VR ₃,₂= 3•3 = 9 a b c
a
a b c
aa ab ac
b
a b c
ba bb bc
c
a b c
ca cb cc
PERMUTACIONES ORDINARIAS O SIN REPETICIÓN Son los diferentes grupos de m elementos distintos entre sí que se pueden formar de manera que en cada dos grupos el orden de los elementos es distinto Se representa por Pm Y se tiene que: M!=m(m-1)(m-2)…3.2.1
PERMUTACIONES CIRCULARES • Se fija un elemento y se hace permutar el resto. PCm=(m-1)!
COMBINACIONES ORDINARIAS O SIN REPITICIÓN • Las combinaciones ordinarias o sin repetición de m elementos tomados de p en p, siendo p≤m, son los diferentes grupos de p elementos distintos entre sí que se pueden formar con los m elementos de forma que en cada dos grupos: • alguno de los elementos es distinto. Se representan por Cm,p, o bien C, y se tiene que: Cm,p = () Para formar las combinaciones ordinarias, se diseña un árbol; en la 1ª columna se colocan todos los elementos, en la 2ª columna por cada elemento de la 1ª columna se colocan todos los elementos que le siguen en orden, etc.
Solving problems •
•
When solving these type of problems is very important the order of the elements, if in each group can include all the elements and if they van be repeated. It is classified with this method
1. Order influences? 1.1 Yes: do all the elements enter in each group? 1.2 No: combinations 1.1.1 No: variations 1.1.2 Yes: permutations
Sample space • The sample space associated with a random simple experiment is formed by the set of all the results that they can present. • Ex: E{1,3,5,7,9} • Elemental event: each one of the results in the sammple space • Event: it is formed by the range of elemental events • Sure event: It’s the only event that can occur • Imposible event:It cannot occur
Events’ operations • Opposite event: of event A, is formed all the elemental events that are not in A. It is represented by A. • Event union: it’s formed by all the elemental events of A and B. It’s represented by AUB • Event intersection: it’s formed by all the elemental events common of A and B. It’s represented by AB
Ley de los grandes números – Large numbers law • Es un teorema en probabilidades que describe el comportamiento del promedio de una sucesión de variables aleatorias según el número total de variables aumenta.
• It states that the probability of an event is the constant which approximates the relative frequency when the experiment is repeated many times.
Regla de Laplace – Rule of succession • P(A) = Nº de casos favorables al suceso A / Nº de casos posibles Lanzamos un dado y queremos calcular la probabilidad de los sucesos A = «Que salga número par» y B = «Que salga número menor que 3». 1. Descomponemos en sucesos elementales tanto el espacio muestral como los sucesos de los que queremos calcular su probabilidad. 2. Contamos el número de sucesos elementales de cada uno y, aplicando la regla de Laplace, calculamos las probabilidades pedidas.
Solución al problema 1.- E = {1, 2, 3, 4, 5, 6} A = {2, 4, 6} B = {1, 2} 2.- P ( A ) = n.° de casos favorables a A / n.° de casos posibles = 3/6 = 1/2 P ( B ) = n.° de casos favorables a B / n.° de casos posibles = 2/6 = 1/3 Virtual Dice
Propiedades de la probabilidad – Properties of probability P(Ā) = 1 – P(A) P(Ø) = 0 P(E) = 1 P(AUB) = P(A) + P(B): incompatibles P(AUB) = P(A) + P(B) - P(A∩B): compatibles 0 ≤ P(A) ≤ 1
Axiomas de la probabilidad
Compound experiments: are formed by many simples experiments.
Throw two coins up
DIAGRAMA CARTESIANO • Tiene utilidad en algunos experimentos compuestos formados por dos simples. 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
• Calcular la probabilidad de que, al lanzar 2 dados, la suma sea 5. • P(5)= 4/36 =1/9= 0,11
DEPENDENT AND INDEPENDENT EVENTS • To explain independent events, we´re going to give an example: An urn has 4 red balls and 5 green balls. A ball is drawn, it preserves the color and re-enter, then another ball is drawn. What is the probability that both are red?
P(RR)= P(R)xP(R)= 4/9x4/9= 16/81=0,20
GRﾃ：ICAMENTE
La probabilidad del suceso B condicionado por el suceso A es la probabilidad de que realice B sabiendo que se ha realizado A. Se representa por P(B/A) . De una baraja de 48 cartas se extrae simultรกneamente dos de ellas. Calcular la probabilidad de que: 1 Las dos sean copas. 2Al menos una sea copas. 3Una sea copa y la otra espada.
Regla del producto o de la probabilidad compuesta La regla del producto o de la probabilidad compuesta dice que la probabilidad de un camino es igual al producto de las probabilidades de las ramas que lo forman P(CC)=P(C)路P(C/C)=10:40路9:39=1:4路3:13=3: 52
Regla de la suma o de la probabilidad total La regla de la suma o de la probabilidad total dice que la probabilidad de varios caminos es igual a la suma de las probabilidades de cada uno de los caminos.