A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift. A quality control consultant is to select 6 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 6 workers has the same chance of being selected as does any other group (drawing 6 slips without replacement from among 45). (Enter your answers to four decimal places.)

Required:
a. How many selections result in all 7 workers coming from the day shift?
b. What is the probability that all 7 selected workers will be from the day shift?
c. What is the probability that all 7 selected workers will be from the same shift?
d. What is the probability that at least two different shifts will be represented among the selected workers?
e. What is the probability that at least one of the shifts will be unrepresented in that sample of workers?

Answer:

Step-by-step explanation:

Total number of workers = 20+ 15 + 10 = 45

Suppose 6 workers are selected from 45 workers, the number of ways they can be selected is [tex](^{45}_6)[/tex] ways

If 7 workers are selected from 20 workers on a day shift;

Then, the number of ways to select 7 members from a group of 20 is [tex](^{20}_7)[/tex] ways;

∴

P( all 7 selected workers will be from the day shift) is:

[tex]= \dfrac{(^{20}_7)}{(^{45}_6)}[/tex]

[tex]=\dfrac{ \dfrac{20!}{7!(20-7)!} } { \dfrac{45!}{6!(45-6)!}}[/tex]

= 0.0095

The possible number of ways to select all day shift workers is:

[tex]=(^{20}_{7})[/tex]

The  possible number of ways to select all swing shift workers is:

[tex]=(^{15}_{7})[/tex]

The possible number of ways to select all graveyard shift workers is:

[tex]= ( ^{10}_{7})[/tex]

∴

The probability that all selected workers are from the same shift is:

[tex]= \Bigg [ \dfrac{ (^{20}_7) }{(^{45}_6) } + \dfrac{ (^{15}_7) }{(^{45}_6) } + \dfrac{ (^{10}_7) }{(^{45}_6) } \Bigg][/tex]

[tex]= \begin {bmatrix} \dfrac{ \dfrac{20!} {7!(20-7)!}}{ \dfrac{ 45! }{6!(45-6)!} } + \dfrac{ \dfrac{15!} {7!(15-7)!}}{ \dfrac{ 45! }{6!(45-6)!} } + \dfrac{ \dfrac{10!} {7!(10-7)!}}{ \dfrac{ 45! }{6!(45-6)!} } \end {bmatrix}[/tex]

= 0.0095 + 0.00079 + 0.000015

= 0.010305

P(selected workers are from at least two different shifts)

= 1 - P(selected workers are from 1 shift from any of the 3 shifts)

= 1 - 0.010305

= 0.989695

≅ 0.9897

We can determine the probability that at least one shift is unrepresented through the following ways:

Let assume that:

[tex]X_1 =[/tex] event that day shift is unrepresented

[tex]X_2 =[/tex] event that swing shift is unrepresented

[tex]X_3 =[/tex] event that graveyard shift is unrepresented

Thus; [tex](X_1 \cap X_2)[/tex] = event that all of them are drawn from graveyard shift.

[tex](X_1 \cap X_3) =[/tex] event that all are drawn from swing shift

[tex](X_2 \cap X_3) =[/tex] event that they are all drawn from day shift.

So;

[tex]P(X_1) = \dfrac{(^{25}_{7})} { (^{45}_{6}) } = 0.0590[/tex]

[tex]P(X_2) = \dfrac{(^{30}_{7})} { (^{45}_{6}) } = 0.2499[/tex]

[tex]P(X_3) = \dfrac{(^{35}_{7})} { (^{45}_{6}) } = 0.8256[/tex]

[tex]P(X_1 \cap X_2) = \dfrac{(^{10}_{7})} { (^{45}_{6}) } = 0.000015[/tex]

[tex]P(X_1 \cap X_3) = \dfrac{(^{15}_{7})} { (^{45}_{6}) } = 0.00079[/tex]

[tex]P(X_2 \cap X_3) = \dfrac{(^{20}_{7})} { (^{45}_{6}) } = 0.0095[/tex]

[tex]P(X_1 \cap X_2 \cap X_3) =0[/tex]

Probability (that at least one shift is unrepresented) is:

[tex]= P(X_1) + P(X_2) +P(X_3) - P(X_1 \cap X_2) - P(X_1 \cap X_3) - P(X_2 \cap X_3) + P(X_1 \cap X_2 \cap X_3)[/tex]

= 0.0590 + 0.2499 + 0.8256 - 0.000015 - 0.00079 - 0.0095 + 0

≅ 1.124

Pablo shares 525 cupcakes in the ratio of 3 : 4 between his bakery and a supermarket.

How many cupcakes does the supermarket get?

What is the inverse function of f(x) = x

Please help don’t understand

7/8 divided by 1/16 in simplest form?

Which of the following can a seismograph not show scientists?

PLEASE ANSWER ASAP !!!!!!!!!!!!!!!!!!! WILL GET BRAINLEST IF CORRECT!!!!!!!!!!!!!!!!!!!!!!

Given the equation y
- x+1
The x intercept is:
The y intercept is:

pls awnser asap plplplplplplplplplplplplp

The width of a rectangle is 7 inches less than the length. The perimeter is 62 inches. Find the length and the width.

The graph of a linear function is shown on the grid . What is the slope of the line

A) -10
B) -3/2
C) -2/3
D) 2​

Who is the protagonist?

A. the magician
b. Aladdin
c. the princess
D. the tailor

Which expressions are solutions to the equation 3/4x=15

Evaluation 6ab when a = 1/2 and b = 7

find the missing value in the ration table

PETS Each month Travis spends $20 for premium food$10 for treats $18 for grooming, and $3 for toys for his pet. Write an expression that could be used to determine the amount he spends on these items for his pet over a period of 12 months.

A cliff diver jumps from a height of 58 feet with an initial velocity of 4 feet per second.

Mr. Santor cuts 1/2 of a piece of construction paper. He uses 1/6 of the piece to make a flower. What fraction of the sheet of paper does he use to make the flower?

Simplify the expression.
- 2.6f +0.2f - 15 - 3
i need help with this, anyone?

A particle traveled a distance of 9.0×1012 meters at a rate of 1.8×105 meters per second. If distance = rate x time, how many seconds did it take the particle to travel the specified distance?

HELP MIDDLE SCHOOL QUESTION

find the equations of the perpendicular line y=2x+1 and passes through (0,-2)

Researchers for the National Institute of Child Health and Human Development studied 208 infants whose brains were temporarily deprived of oxygen as a result of complications at birth. These babies were subjects in an experiment to determine if reducing body temperature for three days after birth improved their chances of surviving without brain damage. What is the population

Scottie is testing to see if people who go to a lot of movies do better than average on the Verbal SAT (which has a mean of 500 and a standard deviation of 100 and is normally distributed). She has a sample of six people. What kind of hypothesis testing do we use

I need help
are all intergers a closed or open whoel number when subtracting or adding them together

imma look what dis is

Explain whether the following statement is true or false. The function notation f-1(x) means 1/f(x)

Rational equations intro (Khan Academy)

Find the perimeter of the following shape:

There are 36 students on the debate team at a school. There are 6 seniors, 12 juniors,
10 sophomores, and 8 freshmen on the team. The coach is going to select 4 students at random to participate in
the next debate. Calculate the probability that all 4 students are sophomores.
O 0.0036
O 0.0060
O 0.0084
O 0.2778

Find the vertex of the parabola defined by the following equation. 3x^2-12x+11