33092626-Ej-Probabilidad-y-as.pdf

ORDENACIONES Y DISTRIBUCIONES DE FRECUENCIAS

1. Ponerlos Ponerlos números números 17, 17, 45, 38, 27, 6, 48, 11, 11, 57, 34, 22, en orden y determinar el Rango 2. Colocar Colocar los números 12, 56, 42, 42, 21, 5, 18, 10, 3, 61, 34, 34, 65, 24, en orden y determinar el Rango 3. Anotar Anotar el peso y estatura estatura de cada uno de sus compañ compañeros eros de clase y ordenarlos de mayor a menor. 4. La puntuación puntuación final en Matemáticas Matemáticas de 80 80 estudiantes estudiantes en la Universidad X se registra en la tabla adjunta 68 73 61 66 96 79 65 86

84 79 65 78 78 62 80 67

75 68 75 82 89 67 73 73

82 73 87 75 61 97 57 81

68 60 74 94 75 78 88 72

90 93 62 77 95 85 78 63

62 71 88 59 78 74 79 65

76 95 69 60 76 62 75 76

85 63 68 83 71 53 76 85

93 75 72 60 71 75 74 77

Con relación a esta tabla, encontrar: La puntuación mas alta • La puntuación mas baja • El rango • Las Las punt puntua uaci cion ones es de los 5 estu estudi dian ante tes s de mayo mayorr • puntuación Las Las punt puntua uaci cion ones es de los 5 estu estudi dian ante tes s de meno menorr • puntuación La puntu ntuación del déc décimo est estudi udiante ante de may mayor • puntuación Cuantos Cuantos estudia estudiantes ntes obtuvier obtuvieron on puntuaci puntuación ón de 75 o • mayor Cuantos estudiantes obtuvieron puntuación menor de • 85 1

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•

•

Que por porcenta ntaje de estudiantes obtuvo una puntuación mayor que 65 pero no mayor que 85 Que puntua puntuaci cione ones s no tienen ningún estudiante

DURACIÓN (HRS) 300 400 500 600 700 800 900 1000 1100

– – – – – – – – –

399 499 599 699 799 899 999 1099 1199

NUMERO DE TUBOS 14 46 58 76 68 62 48 22 6

TOTAL 400 5. La tabla siguiente muestra una distribución de frecuencias de la duración de 400 focos producidos por la compañía X. Con referencia a esta tabla determinar: • • • • • • • •

•

•

Limite superior de la 5ta clase Limite inferior de la 8va clase Marca de clase de la 7ma clase Limites reales de la ultima clase  Tamaño del intervalo de clase Frecuencia de la 4ta clase Frecuencia relativa de la 6ta clase Porcentaje de focos cuya duración no sobrepasa las 600 hrs Porcentaje de tubos cuya duración es mayor o igual a 900 hrs Porcentaje de tubos cuya duración es al menos de 500 hrs pero menor de 1000 hrs

6. La tabla siguiente siguiente muestra muestra una distribuc distribución ión de frecuenc frecuencia ia de los salarios semanales en dólares de 65 empleados de la compañía X. Con referencia a esta tabla determinar:

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• • • • • • • • •

•

El limite inferior de la 6ta clase El limite superior de la 4ta clase La marca de clase ( punto medio) de la 3ra clase Los limites reales de la 5ta clase  Tamaño del 5to intervalo de clase Frecuencia de la 3ra clase Frecuencia relativa de la 3ra clase Intervalo de clase que tiene mayor frecuencia Porcentaje de empleados con salarios menores a 80 por semana Porcentaje de empleado con menos de 100 pero con 60 semanales al menos

7. Constru Construir ir un Histogr Histograma ama y un Polígono Polígono de frecuenc frecuencias ias para el problema anterior 8. La tabla siguie siguiente nte muestra muestra los pesos de 40 estudiante estudiantes s en la Universidad X se registran con aproximación de 1 libra, construir una distribución de frecuencias 138 146 168 146 161

164 158 126 173 145

150 140 138 142 135

132 147 176 147 142

144 125 149 157 136 148 152 144 163 119 154 165 13S 5ALA 15 1E 35 R3IOS140 NUM RO DE 150 156 145 128 EMPLEADOS 50 – 50.99 8 9. Cons Constr trui uirr un hist histog ogra rama ma y 60 – 69.99 10 un polígo polígono no de frecuen frecuencia cias s 70 – 79.99 16 para la distribución de 80 – 89.99 14 frec frecue uenc nciias del del prob proble lema ma 90 – 99.99 10 anterior 100 – 109.99 5 110 – 119.99 2 10. La siguiente tabla TOTAL 65 mues muestr tra a los diám diámet etro ros s en

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0.738 0.728 0.745 0.733 0.735 0.732

0.729 0.737 0.736 0.730 0.732 0.737

0.743 0.736 0.742 0.732 0.735 0.731

0.740 0.735 0.740 0.730 0.727 0.746

0.736 0.724 0.728 0.739 0.734 0.735

0.740 0.733 0.738 0.734 0.732 0.735

0.735 0.742 0.725 0.738 0.736 0.729

0.731 0.736 0.733 0.739 0.741 0.744

0.726 0.739 0.734 0.727 0.736 0.730

0.737 0.735 0.732 0.735 0.744 0.740

Construir Construir una distribución distribución de frecuencias, frecuencias, un Histograma Histograma y un polígono de frecuencias. 11. 11. Cons Conside iderem remos os ahora ahora las sigui siguient entes es 80 medic medicio iones nes de la emisión emisión diaria (en toneladas) toneladas) de oxido de azufre de una planta industrial: 15.8 22.7 9.8 6.2 14.7 17.5 26.1 12.8

26.4 28.6 17.6 23.7 26.8 22.7 18.0 20.5

17.3 11.5 20.9 15.5 19.4 16.7 10.7 19.1

11.2 15.2 2.9 26.6 20.4 21.4 19.2 21.6

23.9 16.9 19.0 18.5 23.0 24.6 20.1 16.2

24.8 18.0 14.4 24.3 22.5 24.6 18.4 18.1

18.7 7.7 29.6 8.3 13.3 11.8 19.3 20.0

13.9 13.5 19.4 21.9 25.7 10.5 9.4 24.1

9.0 23.5 17.0 12.3 31.8 15.9 25.5 201

13.2 14.5 20.8 22.3 25.9 18.1 17.9 28.5

Construir Construir una distribución distribución de frecuencias, frecuencias, un Histograma Histograma y un polígono de frecuencias. 12. 12. Los Los sigui siguient entes es datos datos indi indica can n el numero numero de trabaj trabajado adores res que faltan a una fabrica en 50 días de trabajo 13 3 8 11 29

5 11 21 3 12

13 19 12 6 9

37 6 11 10 19

10 15 7 4 8

16 10 7 6 20

2 14 9 32 15

11 10 16 9 5

6 7 49 12 17

12 24 18 7 10

Construir Construir una distribución distribución de frecuencias, frecuencias, un Histograma Histograma y un

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13. 13. Un expe expert rto o en compu omputa tado dora ras s trat tratan ando do de optim ptimiz izar ar la operación de un sistema, reunió datos sobre el tiempo en microsegundos, entre las solicitudes de un proceso especial 2808 4201 3848

9112

2082

5913

1620

6719

2165 7 3072 2949 1176 4731 1421 1583 9853 7881 6655 8 1 1 1803 7012 1892 4227 6583 1514 4740 8528 1056 7 3 4300 1672 2613 2646 3486 4191 4030 2472 2884 3 3 3 7 0 2448 1400 1524 1643 5732 5419 2860 2487 995 7 1 1 8 3116 2950 1144 2833 3440 8 0 6 Construir Construir una distribución distribución de frecuencias, frecuencias, un Histograma Histograma y un polígono de frecuencias. 14. 14. Se midió midió la resis resiste tenc ncia ia a la comp compre resi sión ón de 58 mues muestr tras as de una aleación de aluminio en desarrollo como material para aeronaves 66.4 69.2 69.6 71.7

67.7 69.3 71.8 70.1

68.0 69.3 70.6 72.3

68.0 69.5 72.1 73.1

68.3 69.5 71.8 70.4

68.4 70.3 72.4 74.5

68.6 71.2 70.0 69.9

68.8 71.3 73.3 73.5

68.9 70.5 70.9 71.6

69.0 71.5 69.8 74.2

69.1 72.2 72.6 71.0

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Construir Construir una distribución distribución de frecuencias, frecuencias, un Histograma Histograma y un polígono de frecuencias. 15. Los siguien siguientes tes datos datos son son medici mediciones ones de la resist resistenc encia ia a la la ruptura (en oz) de una muestra de 60 hilos de cáñamo 32.5 21.2 27.3 20.6 25.4 36.9

15.2 28.3 33.7 29.5 34.1 24.6

35.4 27.1 29.4 21.8 27.5 28.9

21.3 25.0 21.9 37.5 29.6 24.8

28.4 32.7 29.3 33.5 22.2 28.1

26.9 29.5 17.3 29.6 22.7 25.4

34.6 30.2 29.0 26.8 31.3 34.5

29.3 23.9 36.8 28.7 33.2 23.6

24.5 23.0 29.2 34.8 37.0 38.4

31.0 26.4 23.5 18.6 28.3 24.0

Construir Construir una distribución distribución de frecuencias, frecuencias, un Histograma Histograma y un polígono de frecuencias. 16. Los siguie uientes ntes datos son los tiempos pos de ignic nición de ciertos materiales de tapicería expuestos al fuego, dados a la mas cercana centésima de segundo: 2.58 4.79 5.50 6.75 5.84 8.80 7.40 4.72 3.62 2.46

2.51 6.20 5.92 8.75 2.65 7.86 4.71 6.25 9.45 12.80

4.04 1.52 4.56 1.42 1.92 7.60 8.79 5.92 9.65 5.09

6.43 1.38 2.46 4.11 3.81 3.78 3.75 3.10 2.50 1.20

1.58 3.87 6.90 6.37 3.24 6.40 1.70 6.43 5.21 6.85

4.32 4.54 1.47 5.40 1.79 6.77 5.62 9.70 1.76 2.80

2.20 5.12 2.11 11.25 4.90 3.49 4.50 5.11 9.20 7.35

4.19 5.15 2.32 3.90 5.33 8.64 7.41 7.95 10.60 11.75

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17. En un estudio estudio de 2 semanas semanas sobre sobre la produ producti ctivid vidad ad de los trabajadores, se obtuvieron los siguientes datos sobre el numero numero total total de piez piezas as acept aceptabl ables es que produ produje jeron ron 100 100 trabajadores 65 36 49 84 79 56 28 43 67 36

43 78 37 40 68 72 55 62 22 82

59 48 76 74 70 51 40 75 56 45

35 62 52 63 32 80 64 53 74 34

88 50 60 56 57 46 39 57 73 65

76 60 48 55 51 54 45 44 35 51

21 33 61 77 60 85 68 45 53 34

35 67 42 69 52 68 52 47 62 65

61 55 59 41 54 41 74 82 58 26

45 61 73 50 53 35 47 50 38 70

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0.883

0.504

0.667

0.076

0.115

0.277

0.883

0.533

0.084

0.429

0.167

0.938

0.642

0.983

0.687

0.661

0.217

0.270

0.132

0.729

0.276

0.013

0.503

0.470

0.255

0.188

0.530

0.076

0.165

0.683

0.907

0.078

0.560

0.860

0.018

0.591

0.293

0.514

0.964

0.495

0.587

0.738

0.449

0.898

0.027

0.357

0.431

0.813

0.912

0.831

0.623

0.193

0.824

0.109

0.404

0.261

0.069

0.546

0.180

0.018

0.532

0.562

0.096

0.911

0.096

0.482

0.342

0.345

0.343

0.339

0.798

0.239

0.293

0.725

0.192

0.102

0.400

0.05

0.09

0.89

0.28

0.93

0.30

0.49

0.52

0.72

0.84

0.24

0.30

0.65

0.86

0.08

0.11

0.17

0.51

0.36

0.90

0.06

0.28

0.99

0.69

0.65

0.04

0.95

0.03

0.35

0.38

0.05

0.88

0.30

0.55

0.26

0.76

0.11

0.20

0.57

0.90

0.27

0.35

0.06

0.05

0.36

0.49

0.83

0.63

0.99

0.71

0.95

0.96

0.06

0.74

0.29

0.14

0.44

0.09

0.93

0.79

0.72

0.88

0.57

0.72

0.21

0.19

0.22

0.55

0.11

0.27

0.87

0.86

0.21

0.47

0.83

0.78

0.58

0.89

0.62

0.04

0.39

0.81

0.93

0.21

b)

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DIAGRAMAS DE VENN EULER

1. Dado Dados s los los sigu siguie ient ntes es conju conjunt ntos os:: A = {2,4 {2,4,6 ,6,8 ,8,1 ,10} 0},, B= {0,1,2,3 }, C = { -2,-1, 0,3} construye los diagramas de Venn-Euler de AUB AUC • BUC • 2. Dado Dados s los los sigu siguie ient ntes es conj onjunto untos: s: A = {2,4,6 ,4,6,8 ,8,1 ,10} 0},, B ={0,1,2,3 }, C = { -1,-2, 0,3} construye los diagramas de Venn-Euler de •

• • •

A∩B A∩C B∩C

3. Dado Dados s los los sigu siguie ient ntes es conju conjunt ntos os::

A= {2,4, {2,4,6, 6,8, 8,10 10}, }, B=

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•

 A ∪ B

•

 A ∩ B

•

 A c

•

 B

• •

c

( A ∪ B ) c ( A ∩ B ) c

•

\  A \  B

•

( A ∩ B )  B c

•

A  B c

\

6. Sombrear en cada caso lo lo que se se pida: •

( A ∪ B ) ∩ C  ( A ∩ B ) c  A

c

( A ∩ B ∩ C ) c ( A ∪ B ) c ∩ C  ( A ∩ B ) c ∪ A

\ ( B

∪ C 

( A ∩ C  )

\ B

 A

c

c c

( A ∩ B )

\

c

C 

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MEDIDAS DE CENTRALIZACION CENTRALIZACION

1.

Las calificaciones de un estudiante en 6 exámenes fueron, 84, 91, 72, 68, 87 y 78. Hallar la media aritmética, la mediana y la moda de las calificaciones.

2.

Los sal sala ario rios anua anualles de 4 homb ombres res fue fueron ron $ 50, 50,000, $ 60,000, $ 65,000 y $ 30,000, ¿ hallar la media aritmética de estos salarios y se diría que este promedio es representativo de los salarios?

3.

Hall Hallar ar la la med media ia ari aritm tmét étic ica a de los los núm númer eros os 5, 3, 6, 5, 4, 5, 2, 8, 6, 5, 4, 8, 3, 4, 5, 4, 8, 2, 5, 4

4.

De un tot total de 10 100 núm números, 20 er eran 4, 40 40 era eran 5, 30 30 eran 6 y el resto eran 7, hallar la media aritmética de los

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7.

Hallar La media mediana y moda de los siguientes conjuntos de números: 3, 5, 2, 6, 5, 9, 5, 2, 8, 6 • 51.6, 48.7, 50. 3, 49.5, 48.9 • 19.7 19.7,, 21.5 21.5,, 22.5 22.5,, 22.2 22.2,, 22.6 22.6,, 21.9 21.9,, 20.5 20.5,, 19.3 19.3,, 19.9 19.9,, • 21.7 21.7,, 22.8 22.8,, 23.2 23.2,, 21.4 21.4,, 20.8 20.8,, 19.4 19.4,, 22.0 22.0,, 23.0 23.0,, 21.1 21.1,, 20.9, 21.3

8.

Por Por erro errorr un pro profe feso sorr borr borro o la la cali califi fica caci ción ón que que obt obtuv uvo o uno uno de sus 10 alumnos, si los otros 9 consiguieron las calificaciones de 43, 66, 74, 90, 40, 52, 70, 78 y 92 y si la media de los 10 estudiantes es de 67 ¿ Cual es la calificación que borro el profesor? 9. Los si sigui guientes ntes dat datos os son son el el num numero ero de min minutos utos que que en en 15 días laborales una persona tiene que esperar el autobús que la llevara a su trabajo: 10, 1, 13, 9, 5, 9, 2, 10, 3, 8, 6, 17, 2, 10 y 15, encuentre la media, mediana y moda.

10. 10. Encu Encuen entr tre e la medi media, a, medi median ana a y moda moda para para el sigui siguien ente te conjunto de números

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0.893

0.197

0.702

0.213

0.714

11. 11. Obte Obtene nerr la media media,, media mediana na y moda moda del prob proble lema ma de los los pesos y las estaturas.

MEDIDAS DE DISPERCION

1. Hallar Hallar la desviació desviación n media media de los siguien siguientes tes conjunto conjuntos s de números : • •

2, 3, 6, 8, 11 12.6, 7.3, 15, 10, 18.5

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5. Encont Encontra rarr la media media,, median mediana, a, moda, moda, desvi desviac ació ión n media media y desviación estándar para datos agrupados de las 13 tablas de la sección de Distribuciones de frecuencia.

TEORIA ELEMENTAL DE LA PROBABILIDAD

1. Determin Determinar ar La probabi probabilid lidad ad p o un estimad estimador or de ella ella para cada uno de los siguientes sucesos:

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•

•

•

•

La aparición en una sola tirada de 2 dados de una suma de 8 puntos Un cerrojo no defectuoso a extraer de una población si de 600 ya examinados, 12 fueron defectuosos La aparición en una sola tirada de dos dados de la suma 7 u 11 la aparición de al menos de una cara en tres lanzamientos de una moneda

2. Cier Cierta tas s prue prueba bas s mues muestr tran an que que 294 294 de 300 300 aisl aislan ante tes s de cerá cerámi mica ca proba probado dos s podrí podrían an resi resist stir ir un choqu choque e térmi térmico co,, ¿ Cual Cual es la prob probab abil iliidad dad de que que cual cualqu quie iera ra de tale tales s aislantes pueda resistirlo ? 3. Si 3 de 20 neumático neumáticos s están defectuos defectuos

tro de ellos

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6.

Un experimento consiste consiste en tirar un dado y una moneda, si  E  es el suceso cara en la moneda y  E  es el suceso 3 o 6 en el dado que significa: 2

1

•

 E 1

•

 E 2

•

 E 1

• •

+ E 2

 P { E 1 E 2 }  p{ E 1

+

E 2 }

7. Se saca al azar una bola bola de una caja caja que contiene contiene 6 rojas, rojas, 4 blanc blancas as,, 5 azul azules es,, hall hallar ar la proba probabi bili lidad dad de que la bola bola extraída sea: • • • •

Roja Blanca Azul No roja

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10. 10. Hall Hallar ar la la proba probabi bili lidad dad de sal salir ir exen exento to 11. 11. Hall Hallar ar la prob probab abil ilid idad ad de sali salirr exen exento to en cada cada ocasi ocasión ón después de cada extracción 12. 12. Se hacen hacen 2 extr extrac acc ciones ones de una una bara baraja ja de 52 cart cartas as,, hallar la probabilidad de que las 2 cartas extraídas sean ases, siendo las extracciones : • •

Con emplazamiento Sin emplazamiento

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•

La segunda carta no sea figura dado que la primera lo fue

16. 16. Hall Halla ar la prob proba abil bilidad idad de obten btener er al meno menos s un 4 en 2 lanzamientos de un dado 17. Una bols bolsa a contie contiene ne 4 bolas bolas blanca blancas s y 2 bola bolas s negras negras,, otra otra contiene 3 bolas blancas y 5 bolas negras, si se extrae una bola de cada bolsa hallar la probabilidad de que • •

Ambas sean blancas Ambas sean negras

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6. Se quier quieren en senta sentarr 5 homb hombre res s y 4 muje mujere res s en una fila fila de modo que las mujeres ocupen los sitios pares ¿ de cuantas formas pueden sentarse? 7. ¿Cuánto ¿Cuántos s números números de 4 cifras cifras pueden formars formarse e con los 10 dígitos 0,1, 2, 3, ......, 9 si los números pueden repetirse y • • • •

No hay restricción El primer numero es impar Los números son pares Los números son impares El primer numero es par

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2 personas determinadas no deben estar una al lado de la otra

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5. Con 7 conson nsona antes ntes y 5 vocales diferent rente es ¿ cuanta antas s palabras pueden formarse que consten de 4 consonantes y 3 vocales ?. Se admiten palabras sin significado. 6.

Para que valor de n se cumple que

3⋅

n +1

C 3

=7⋅ n

C 2

7. De cuantas formas formas pueden pueden seleccionarse seleccionarse 6 preguntas preguntas de un total de 10 8. Cuan Cuanto tos s comit comités és dife difere rent ntes es de 3 homb hombre res s y 4 mujer mujeres es pueden formarse con 8 hombres y 6 mujeres

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• • • •

Las 3 sean rojas Las 3 sean blancas 2 sean rojas y una blanca al menos una sea blanca

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• • •

Ninguna vez Una vez Dos veces

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8. La proba probabil bilid idad ad de que un estudi estudiant ante e se gradué gradué en cier cierta ta universidad es 0.4. Determinar la probabilidad de que de 5 estudiantes nuevos se gradué

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una tercera parte. De acuerdo con lo anterior, ¿ cuales son las probabilidades de que se reduzcan al menos en una tercera parte en

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19. 19. Supó Supóng ngas ase e que que un exame examen n en la admis admisió ión n publ public ica a esta esta diseñado en forma tal que el 70% de las personas con un CI de 90 lo aprueben. Encontrar las probabilidades de que

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