a) Select ( 5 Day workers ) = 252, P ( 5 Day Workers ) = 0.006
b) P ( 5 same shift ) = 0.0074612
c) P ( At-least 2 different shifts ) = 0.9925
d) P ( Only 2 different shifts ) = 0.3366
Step-by-step explanation:
Given:-
- The number of day shift workers D = 10
- The number of swing shift workers S = 8
- The number of graveyard shift workers G = 6
- The total selection made by the Quality Team = 5
Find:-
(a) How many selections result in all 5 workers coming from the day shift?What is the probability that all 5 selected workers will be from the day shift?
Solution:-
- To select the slips such that all 5 are for Day shift workers, in other words you are also selecting 5 Day shift workers from a pool of 10 Day shift workers. The following number of combinations would be:
Select ( 5 Day workers ) = 10 C 5 = 252 combinations
- The total possible outcomes for selecting 5 workers from any of the shifts is:
Select ( 5 Workers ) = 24 C 5 = 42504 combinations
- The associated probability for selecting 5 day shift workers is:
P ( 5 Day Workers ) = Select ( 5 Day workers ) / Select ( 5 Workers )
= 252 / 42504
= 0.006
Find:-
(b) What is the probability that all 5 selected workers will be from the same shift? (Round your answer to four decimal places.)
Solution:-
- To select the slips such that all 5 are from same shift, in other words you are selecting 5 Day shift workers, or 5 Swing shift workers or 5 Graveyard shift workers from a pool of 10 Day shift workers, 8 Swing shift workers, 6 graveyard shift workers. The following number of combinations would be:
Select ( 5 Day workers ) = 10 C 5 = 252 combinations
Select ( 5 Swing workers ) = 8 C 5 = 56 combinations
Select ( 5 Graveyard workers ) = 6 C 5 = 6 combinations
- The associated probability for selecting 5 same shift workers is:
P ( 5 Day Workers ) = Select ( 5 Day workers ) / Select ( 5 Workers )
= 252 / 42504
= 0.006
P ( 5 Swing Workers ) = Select ( 5 swing workers ) / Select ( 5 Workers )
= 56 / 42504
= 0.00132
P ( 5 Graveyard Workers ) = Select ( 5 graveyard workers ) / Select ( 5 Workers )
= 6 / 42504
= 0.0001412
- P ( 5 same shift ) = P ( 5 Day Workers ) + P ( 5 Swing Workers ) + P ( 5 Graveyard Workers )
= 0.006 + 0.00132 + 0.0001412
= 0.0074612
Find:-
(c) What is the probability that at least two different shifts will be represented among the selected workers?
Solution:-
- To select the slips such that all 5 are from different shifts, in other words you are selecting either a combination of Day shift workers and Swing shift workers or Day shift and Graveyard shift workers or Swing shift and Graveyard shift workers or a combination of all 3. It would be easier if we subtract the probability of no different workers from 1 to get at-least 2 different workers probability. As follows:
- P ( At-least 2 different shifts ) = 1 - P ( 5 same shift Workers )
= 1 - 0.0074612
= 0.9925
Find:-
(d) What is the probability that at least one of the shifts will be unrepresented in the sample of workers?
Solution:-
- To select the slips such that all 5 are from only 2 different shifts, in other words you are selecting either a combination of Day shift workers and Swing shift workers or Day shift and Graveyard shift workers or Swing shift and Graveyard shift workers but not a combination of all 3. It would be easier if we subtract the probability of all different workers from the probability of at-least 2 different workers. As follows:
- P ( Only 2 different shifts ) = P ( At-least 2 different ) - P ( All 3 shift Workers )
- The possible combinations for all 3 different shift workers is:
Select ( 3 D , 1 S , 1 G ) = 10 C 3 * 8 * 6 = 5,760
Select ( 2 D , 2 S , 1 G ) = 10 C 2 * 8 C 2 * 6 = 7,560
Select ( 2 D , 1 S , 2 G ) = 10 C 2 * 8 * 6 C 2 = 5,400
Select ( 1 D , 2 S , 2 G ) = 10 * 8 C 2 * 6C2 = 4,200
Select ( 1 D , 3 S , 1 G ) = 10 * 8 C 3 * 6 = 3,360
Select ( 1 D , 1 S , 3 G ) = 10 * 8 * 6 C 3 = 1,600
Total All 3 different shifts selected = 27,880
P ( All 3 shift Workers ) = 27,880 / 42504 = 0.655938264
- Hence,
P ( Only 2 different shifts ) = 0.9925 - 0.655938264
= 0.3366
