Question

A production facility ernploys 10 workers on he day shift, 8 workers on the swing shift, and 6 workers on the graveyard shift. A qualily control consullant 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? selections 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 differhifs will be resented among the selected workers? (Round your answer to four decimal places.) (d) What ls the probability that at lcast one of the shifts wlill be unrepresented in the sample of workers? (Round your answer to four declmal places.)

Answer 1

Solution : ( a )

Number of selections result in all 4 workers coming from the day shift

$=\mathrm{\frac{10C_{4}*8C_{0}* 6C_{0}}{24C_{4}}}$

$=\frac{210*1*1}{10626}$

$=\frac{210}{10626}$

$=\frac{5}{253}$

$=0.0198$

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Solution : ( b )

Probability that all 4 selected workers will be from the same shift

$=\mathrm{\frac{10C_{4}+8C_{4}+6C_{4}}{24C_{4}}}$

$=\frac{210+70+15}{10626}$

$=\frac{295}{10626}$

$=0.0277$

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Solution : ( c )

Probability at least 2 different shifts will be represented among the selected workers

= 1- ( Probability that all 4 selected workers will be from the same shift )

$= 1 - \left ( \mathrm{\frac{10C_{4}+8C_{4}+ 6C_{4}}{24C_{4}}} \right )$

$= 1 - \left ( \frac{210+70+15}{10626} \right )$

$= 1 - \left ( \frac{295}{10626} \right )$

$= 1 - ( 0.0277 )$

$=0.9723$

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Solution : ( d )

Probability that at least 1 of the shifts will be unrepresented in the sample of workers

$= \mathrm{\frac{18C_{4}+14C_{4}+16C_{4}-10C_{4}-8C_{4}-6C_{4}}{24C_{4}}}$

$= \frac{3060+1001+1820-210-70-15}{10626}$

$=\frac{5586}{10626}$

$=\frac{133*42}{253*42}$

$\mathrm{Cancel\:the\:common\:factor:}\:42$

$=\frac{133}{253}$

$=0.5257$