Mathematics, 20.06.2020 16:57 uhhgray

A production facility employs 10 workers on the day shift, 8 workers on the swing shift, and 6 workers on the graveyard shift. A quality control consultant is to select 4 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 4 workers has the same chance of being selected as does any other group (drawing 4 slips without replacement from among 24). (a) How many selections result in all 4 workers coming from the day shift? What is the probability that all 4 selected workers will be from the day shift? (Round your answer to four decimal places.)
(b) What is the probability that all 4 selected workers will be from the same shift? (Round your answer to four decimal places.)
(c) What is the probability that at least two different shifts will be represented among the selected workers? (Round your answer to four decimal places.)
(d) What is the probability that at least one of the shifts will be unrepresented in the sample of workers? (Round your answer to four decimal places.)

Answers: 3

Show answers

Answers

Answer from: rachel2005smith

The probability that all 4 selected workers will be from the day shift is, = 0.0198

The probability that all 4 selected workers will be from the same shift is = 0.0278

The probability that at least two different shifts will be represented among the selected workers is = 0.9722

The probability that at least one of the shifts will be unrepresented in the sample of workers is P(A∪B∪C) = 0.5257

To solve this question properly, we will need to make use of the concept of combination along with set theory.

What is Combination?

In mathematical concept, Combination is the grouping of subsets from a set without taking the order of selection into consideration.

The formula for calculating combination can be expressed as:

$\mathbf{(^n _r) =\dfrac{n!}{r!(n-r)! }}$

From the parameters given:

Workers employed on the day shift = 10Workers on swing shift = 8Workers on graveyard shift = 6

A quality control consultant is to select 4 of these workers for in-depth interviews:

Using the expression for calculating combination:

(a)

The number of selections results in all 4 workers coming from the day shift is :

$\mathbf{(^n _r) = (^{10} _4)}$

$\mathbf{=\dfrac{(10!)}{4!(10-4)!}}$

= 210

The probability that all 5 selected workers will be from the day shift is,

$\begin{array}{c}\\P\left( {{\rm{all \ 4 \ selected \ workers\ will \ be \ from \ the \ day \ shift}}} \right) = \dfrac{{\left( \begin{array}{l}\\10\\\\4\\\end{array} \right)}}{{\left( \begin{array}{l}\\24\\\\4\\\end{array} \right)}}\\\end{array}$

$\mathbf{= \dfrac{210}{10626}} \\ \\ \\ \mathbf{= 0.0198}$

(b) The probability that all 4 selected workers will be from the same shift is calculated as follows:

P( all 4 selected workers will be) $\mathbf{= \dfrac{ \Big(^{10}_4\Big) }{\Big(^{24}_4\Big)}+\dfrac{ \Big(^{8}_4\Big) }{\Big(^{24}_4\Big)} + \dfrac{ \Big(^{6}_4\Big) }{\Big(^{24}_4\Big)}}$

where;

$\mathbf{\Big(^{8}_4\Big) = \dfrac{8!}{4!(8-4)!} = 70}$

$\mathbf{\Big(^{6}_4\Big) = \dfrac{6!}{4!(6-4)!} = 15}$

P( all 4 selected workers is:)

$\mathbf{=\dfrac{210+70+15}{10626}}$

The probability that all 4 selected workers will be from the same shift is = 0.0278

(c)

The probability that at least two different shifts will be represented among the selected workers can be computed as:

$= 1-\dfrac{ (^{10}_4) }{(^{24}_4)}+\dfrac{ (^{8}_4) }{(^{24}_4)} + \dfrac{ (^{6}_4) }{(^{24}_4)}$

$=1 - \dfrac{210+70+15}{10626}$

= 1 - 0.0278

= 0.9722

The probability that at least two different shifts will be represented among the selected workers is = 0.9722

(d)

The probability that at least one of the shifts will be unrepresented in the sample of workers is:

$P(AUBUC) = \dfrac{(^{6+8}_4)}{(^{24}_4)}+ \dfrac{(^{10+6}_4)}{(^{24}_4)}+ \dfrac{(^{10+8}_4)}{(^{24}_4)}- \dfrac{(^{6}_4)}{(^{24}_4)}-\dfrac{(^{8}_4)}{(^{24}_4)}-\dfrac{(^{10}_4)}{(^{24}_4)}+0$

$P(AUBUC) = \dfrac{(^{14}_4)}{(^{24}_4)}+ \dfrac{(^{16}_4)}{(^{24}_4)}+ \dfrac{(^{18}_4)}{(^{24}_4)}- \dfrac{(^{6}_4)}{(^{24}_4)}-\dfrac{(^{8}_4)}{(^{24}_4)}-\dfrac{(^{10}_4)}{(^{24}_4)}+0$

$P(AUBUC) = \dfrac{1001}{10626}+ \dfrac{1820}{10626}+ \dfrac{3060}{10626}-\dfrac{15}{10626}-\dfrac{70}{10626}-\dfrac{210}{10626} +0$

The probability that at least one of the shifts will be unrepresented in the sample of workers is P(A∪B∪C) = 0.5257

Learn more about combination and probability here:

Answer from: Quest

What’s the question?
Max points i can ! it’s in math because i trust the math people more.

Answer from: Quest

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 20:00

Karen will divide her garden into equal parts. she will plant corn in 8/12 of the garden. what is the lowest number of parts she can divide her garden into?

Answers: 1

Answer

Mathematics, 22.06.2019 00:00

Aspacecraft can attain a stable orbit 300 kilometers above earth if it reaches a velocity of 7.7 kilometers per second. the formula for a rocket's maximum velocity v in kilometers per second is vequalsminus0.0098tplusc ln upper r, where t is the firing time in seconds, c is the velocity of the exhaust in kilometers per second, and r is the ratio of the mass of the rocket filled with fuel to the mass of the rocket without fuel. find the velocity of a spacecraft whose booster rocket has a mass ratio of 20, an exhaust velocity of 2.1 km/s, and a firing time of 15 s. can the spacecraft achieve a stable orbit 300 km above earth?

Answers: 3

Answer

Mathematics, 22.06.2019 00:30

The value, v, of kalani’s stock investments over a time period, x, can be determined using the equation v=750(0.80)^-x. what is the rate of increase or decrease associated with this account? 20% decrease 20% increase 25% decrease 25% increase

Answers: 1

Answer

Mathematics, 22.06.2019 02:00

How many square inches are in 60 square feet?

Answers: 1

Answer

You know the right answer?

A production facility employs 10 workers on the day shift, 8 workers on the swing shift, and 6 worke...

Questions

Mathematics, 21.02.2021 14:00

Height of a cone is 30cm, its slant height is 32 cm, then the radius is- ?