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• Juan Sebastian Valbuena Garcia

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Juan Sebastián Valbuena García Taller [4] HYPERGEOMETRIC DISTRIBUTION 1. A small voting district has 101 female voters and 95 male voters. A random sample of 10 voters is drawn. a.) What is the probability exactly 7 of the voters will be female?    

N=196 n=10 x=7 d=101

P ( X=7 )=

( 101C 7 ) [ ( 196−10 ) C(10−7) ] ( 101 C 7 ) ( 95 C 3 ) = ( 196 C 10 ) ( 196 C 10 )

P ( X=7 )=0.1303=13.039 % b.) What is the probability that at least 5 voters will be male?    

N=196 n=10 x≥5 d=95 4

P ( X ≥5 )=∑ i=0

( 95 C X ) [ 101C (10− X) ] ( 196 C 10 )

P ( X ≥5 )=58.72 % c.) What is the probability that there are between 2 and 7 male voters?    

N=196 n=10 2 ≤ x ≤7 d=95 7

P ( 2≤ x ≤ 7 )=∑ i=2

( 95 C X ) [ 101 C(10−X ) ] ( 196 C 10 )

P ( 2≤ x ≤ 7 )=94.74 % 2. A building company has a total of 120 beams in its warehouse. The supplier told the company that the total amount of damaged beams in the order is 23. The quality inspector took a sample of the 10% of the total of the beams in the warehouse. a.) Find the probability of finding 3 damaged beams in the sample taken.

   

N=120 n=12 x=3 d=23 P ( X=3 )=

( 23C 3 )( 97 C 9 ) (120 C 12 )

P ( X=3 )=24.006 %

b.) Find the probability of finding at least 10 damaged beams in the sample taken.    

N=120 n=12 x ≥ 10 d=23 9

P ( X ≥10 )=1−∑ i=0

( 23 C X ) ( 97 C (12− X) ) ( 120 C 12 )

P ( X ≥10 )=5.178∗10−5 %

c.) Find the probability of finding 15 damaged beams in the sample taken.

P ( X=15 )=0 % El resultado es “0” por lo que el valor de la muestra es menor al número de vigas a encontrar

d.) Find the expected amount of damaged beams and its variance.

E [ X ]=

( 12 ) (23) =2.3 (120)

Var [ X ] =

108 (12)(23) 23 1− =1.687 119 120 120

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3. A construction project manager in a construction multinational realizes that the 23% of all the construction project developed by the company has some delay. So, he decides to go to 15 projects from a total of 53 that the company possesses. a.) Find the probability of finding that none of the projects has a delay.    

N=53 n=15 x=0 d=12

P ( X=0 )=

( 12C 0 )( 41C 15 ) ( 53 C 15 )

P ( X=0 )=0.01=1% b.) Find the probability of finding at most 5 projects without delays. 

x≤5 5

P ( x ≤ 5 )= ∑ i=0

( 12C X ) ( 41 C(15−X ) ) ( 53 C 15 )

P ( x ≤ 5 )=93.41 % c.) Find the expected value and the variance of the construction projects with delays.

E [ X ]=

( 15 ) (12) =3 .39 (53)

Var [ X ] =

38 (180) 12 1− =1. 919 52 53 53

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4. In a laboratory there are a total of 70 units of material taken from the soil in a city. You, as a researcher in that laboratory, take a sample of 15 units. The probability of each unit of material taken being damaged is 0,15. a.) Find the probability of not finding any damaged unit of material in the sample is damaged.    

N=70 n=15 x=0 d=11 P ( X=0 )=

( 11 C 0 ) ( 59C 15 ) =5.529 % ( 70C 15 )

b.) Find the probability that there will be between 2 and 7 units of materials damaged. 

2 ≤ x ≤7 7

P ( 2≤ x ≤ 7 )=∑ i=2

( 11 C X ) [ 59 C (15− X) ] =74.18 % ( 70C 15 )

c.) Find the expected value and variance of the units of non-damage units of material.  

d=11→ Material dañado d=59→ Material sin daño

E [ X ]=

885 =12.64 70

Var [ X ] =

55 885 59 1− =1.58 69 70 70

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BINOMIAL DISTRIBUTION 5. In a factory, the probability that a brick produced is defective is 0,78. If you have a warehouse with a total of 200 bricks. a.) Find the probability that there is a total of 17 defective bricks in the warehouse.

P=78 % ( probabilidad de que ocurra) Q=22% ( probabilidad de que no ocurra) n=200 x=17 P ( x=17 )= [ 200 C 17 ] [ 0.7817 ] [ 1−0.78 ] P ( x=17 )=0

183

b.) Find the probability that there are no defective bricks in the warehouse. 200

P ( x=0 )= [ 200 C 0 ] [ 0.780 ] [ 1−0.78 ] P ( x=0 )=0

c.) Find the expected amount of defective bricks in the warehouse.

E [ X ] =156 d.) Find the variance of defective bricks.

Var [ X ] =37.44

6. In a laboratory there are a total of 100 samples of water. The percentage of contaminated water is known to be historically a 23% of all the samples taken. a.) Find the probability that 17 samples of water are contaminated.

n=100 P=0.23 x=17 83

P ( x=17 )= [ 100 C 7 ] [ 0.2317 ] [ 1−0.23 ] =3.55 % b.) Find the probability that at least 8 samples of water are not contaminated.

P=0.77 7

[∑

P ( x ≥ 8 )=1−

100− X

[ 100 C X ] [ 0.77 X ] [ 1−0.77 ]

i=0

]

P ( x ≥ 8 )=100 % c.) Find the probability that no sample of water is contaminated. 100

P ( x=0 )= [ 100 C 0 ] [ 0.230 ] [ 1−0.23 ] =4.45∗10−10 P ( x=0 ) ≅ 0 d.) Find the expected value and variance of the samples of water contaminated.

E [ X ] =23 Var [ X ] =17.71 7. A manager in a construction company realizes that historically 70% of the construction project he manages has at least one accident per month. The company has a total of 27 construction projects. a.) Find the probability that at most 15 projects has at least one accident per month.

n=27 P=0.7 15

P ( x ≤ 15 )=∑ [ 27 C X ] [ 0.7 X ] [ 1−0.7 ]

27− X

=7.98 %

i=0

b.) Find the probability that no project has at least one accident per month. 27

P ( x=0 )= [ 27 C 0 ] [ 0.7 0 ] [ 1−0.7 ] =0 c.) Find the probability that there are between 3 and 8 projects with at least one accident per month. 8

P ( 3≤ x ≤ 8 )=∑ [ 27 C X ] [ 0.7 X ] [ 1−0.7 ]

27−X

=1.783∗10−3 %

i=3

P ( 3≤ x ≤ 8 ) ≅ 0 d.) Find the expected amount of projects without accidents per month.

0.7=Proyectos con accidentes por mes 0.3=Proyectos sin accidentes por mes E [ X ] =( 27 )( 0.3 )=8.1

POISSON DISTRIBUTION 8. Accidentality rate in a company is (according to historical data) is 2 accidents per week. a.) Find the probability that there will be ten accidents in 1 month.

x=10 λ=8 e−8∗810 ( ) P x=10 = =9.92 % 10! b.) Find the probability that there will between 4 and 7 accidents in 3 days.

4 ≤ x ≤7 6 λ= 7 −6 7

P ( 4 ≤ x ≤ 7 ) =∑

6 7

x

( ) =1.14 %

e7∗

x!

i= 4

c.) Find the probability that there will be at least 7 accidents in two and a half months.

P ( x ≥ 7 )=1−P(x< 7) λ=20 6

P ( x ≥ 7 )=1−∑ i=0

e−20∗( 20 )x =99.974 % x!

d.) Find the expected amount of accidents per week.

E [ X ] =( 27 )( 0.3 )=2

9. The frequency in which employees are absent to work in a company is usually 4 absences per month. a.) Find the probability that there will be 5 absences in a week.

x=5 λ=1 P ( x=5 )=

e−1∗15 =0.3 % 5!

b.) Find the probability that there will be no absences in 2 months.

[

4 aus 2 meses → λ=8 1 meses 1

][

P ( x=0 )=

]

e−8∗80 =0. 033 % 0!

c.) Find the probability that there will be at most 10 absences in 3 months.

x≤10 λ=1 2 10

P ( x ≤ 10 )=1−∑ i=0

e−1 2∗( 1 2 )x =34.73 % x!

d.) Find the variance of the amount of accidents per week.

[

4 aus 1 mes

][

1 mes → λ=1 4 semanas

]

NEGATIVE BINOMIAL DISTRIBUTION 10. The quality manager in a company that produces bricks knows that the probability that a brick is defective is 0,13 in the production line. He takes 15 samples to inspect them. a.) Find the probability that the 7th brick chosen randomly for inspection is the third damaged one.

x=7 k =3 4

P ( x=7 )= ( 6C 2 ) ( 0.133 ) ( 1−0.13 ) =1.887 % b.) Find the probability that the 13th brick chosen for inspection is the 7th damaged one.

x=13 k =7 6

P ( x=13 )=( 12 C 6 ) ( 0.137 ) ( 1−0.13 ) =0.025 % c.) Find the expected value and variance for defective bricks. a.) μ=

σ 2=

3 ( 1−0.13 ) =20.07 0.13

3 ( 1−0.13 ) =154.437 0.132

b.) μ=

σ 2=

7 ( 1−0.13 ) =46.84 0.13

( 1−0.13 ) =360.355 0.132

11. In the warehouse of a construction materials supplier the quality assistant takes a sample of 20 beams. Usually 10% of the beams are defective. a.) Find the probability that the 10th sample taken for inspection is the second defective unit found. 8

P ( x=10 )=( 9 C 1 ) ( 0.12 ) ( 1−0.1 ) =3.87 % b.) Find the probability that the 4th sample taken for inspection is the third defective unit found. 1

P ( x=4 )=( 3 C 2 ) ( 0.13 ) ( 1−0.1 ) =0.27 % c.) Find the probability that the seventh sample taken for inspection is the eighth defective unit found.

x=7 k =8 P ( x=7 )= ( 6C 7 ) ( 0.18 ) ( 1−0.1 ) =No hay probabilidad x< k −1

GEOMETRIC DISTRIBUTION 12. In a company there’s a line of beams production, you need to analyze the quality, so take a sample of 10 beams randomly, historically it’s known that the percentage of defective beams is between 8% and 10% out of the total beams produced. a.) Which is the probability that the 6th beam taken as a sample is the first defective.

P=9 % P ( x=6 )=0.09(1−0.09)5=5.616 % b.) Which is the probability that the tenth beam taken as a sample is the first defective.

P ( x=10 )=0.09 (1−0.09)9=0.82 % c.) Find the expected value and variance of defective beams produced.

E [ X ]=

1−0.09 =10.11 0.09

Var [ X ] =

1−0.09 =1 12.345 0.092

13. In a building company there is an analysis process to detect delays in several of their projects. The probability of having a delay in each project is 0,23. a.) Which is the probability that the 5th project taken as a sample is the first in having a delay.

P ( x=5 )=0. 23(1−0. 23) 4=8.085 % b.) Which is the probability that the first project taken as a sample is the first in having a delay.

P ( x=1 )=0.23(1−0.23)0=23 % c.) Which is the probability that the project immediately after the 7 th one is the first one to show delays.

P ( x=8 )=0.23(1−0.23)7=3.7 %