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# Determine the following probabilities.

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1. Four (standard) dice (cubes with 1,2,3,4,5,6 respectively,dots on their six faces) each of a different color are tossed each landing with one of its faces up, thereby showing a number of dots. Determine the following probabilities.

a. the probability that the total number of dots shown is 6
b. the probability that at most two of the dice show exactly one dot
c. the probability that each die shows at least two dots
d. the probability that the four numbers of dots shown are all different
e. the probability that there are exactly two different numbers of dots shown

https://brainmass.com/math/combinatorics/determine-the-following-probabilities-270463

#### Solution Preview

Four dice are tossed and there are 6^4 = 1296 permutations.
a. If the total number of dots is 6, then the possible combinations are (1,1,1,3) and (1,1,2,2), each has 4! = 24
permutations. Thus the probability is (24 + 24) / 1296 = 1/27
b. If at most two of dice show ...

#### Solution Summary

Determine the probabilities.

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## This posting provides solution to several operation management problems like PERT, Probability, Project Schedule, Operating Characteristics etc.

13-17
After consulting with Butch Radner, George Monohan was able to determine the activity times for constructing the weed-harvesting machine to be used on narrow rivers. George would like to determine ES, EF, LS, LF, and slack for each activity. The total project completion time and the critical path should also be determined. (See Problem 13-16 for details). The activity times are shown in the following table:

ACTIVITY TIME (WEEKS)
A 6
B 5
C 3
D 2
E 4
F 6
G 10
H 7

13-20
Using PERT, Ed Rose was able to determine that the expected project completion time for the construction of a pleasure yacht is 21 months and the project variance is 4.
(a) What is the probability that the project will be completed in 17 months or less?
(b) What is the probability that the project will be completed in 20 months or less?
(c) What is the probability that the project will be completed in 23 months or less?
(d) What is the probability that the project will be completed in 25 months or less?

13-21
The air pollution project discussed in the chapter has progressed over the past several weeks, and it is now the end of week 8. Lester Harky would like to know the value of the work completed, the amount of any cost overruns or underruns for the project, and the extend to which the project is ahead of or behind schedule by developing a table like Table 13-8 on page 544. The revised cost figures are shown in the following table:

ACTIVITY PERCENT OF COMPLETION ACTUAL COST (\$)
A 100 20,000
B 100 36,000
C 100 26,000
D 100 44,000
E 50 25,000
F 60 15,000
G 10 5,000
H 10 1,000

13-30. Zuckerman Wiring and Electric is a company that installs wiring and electrical fixtures in residential construction. John Zuckerman has been concerned with the amount of time that it takes to complete wiring jobs. Some of his workers are very unreliable. A list of activities and their optimistic, their pessimistic and their most likely completion times in days are given in the following table:
ACTIVITY DAYS IMMEDIATE
PREDECESSORS
a m b
A 3 6 8
B 2 4 4
C 1 2 3
D 6 7 8 C
E 2 4 6 B, D
F 6 10 14 A, E
G 1 2 4 A, E
H 3 6 9 F
I 10 11 12 G
J 14 16 20 C
K 2 8 10 H, I
Determine the expected completion time and variance for each activity.

13-31. John Zuckerman would like to determine the total project completion time and the critical path for installing electrical wiring and equipment in residential houses. See Problem 13-30 for details. In addition, determine ES, EF, LS, LF, and slack for each activity.

13-32. What is the probability that Zuckerman will finish the project described in Problems 13-30 and 13-31 in 40 days or less?
14-11
The Rockwell Electronics Corporation retains a service crew to repair machine breakdowns that occur on an average of A = 3 per day (approximately Poisson in nature), The crew can service an average of J.l. = 8 machines per day, with a repair time distribution that resembles the exponential distribution.
(a) What is the utilization rate of this service system?
(b) What is the average downtime for a machine that
is broken?
(c) How many machines are waiting to be serviced at any given time?
(d) What is the probability that more than one machine is in the system? Probability that more than two are broken and waiting to be repaired or being serviced? More than three? More than four?

14-19
Juhn and Sons Wholesale Fruit Distributors employ one worker whose Job 1S to load fruit on outgoing company trucks. Trucks arrive at the loading gate at an average of 24 per day, or 3 per hour, according to a Poisson distribution. The worker loads them at a rate of 4 per hour, following approximately the exponential distribution in service times. Determine the operating characteristics of this loading gate problem. What is the probability that there will be more than three trucks either being loaded or waiting? Discuss the results of your queuing model computation.

14-20
Juhn believes that adding a second fruit loader will substantially Improve the firm s efficiency. He estimates that a two-person crew, still acting like a single-server system, at the loading gate will double the loading rate from 4 trucks per hour to 8 trucks per hour. Analyze the effect on the queue of such a change and compare the results with those found in Problem 14-19.

14-21
Truck drivers working for Juhn and Sons (see . Problems 14-19 and 14-20) are paid a salary of \$10 per hour on average. Fruit loaders receive about \$6 per hour. Truck drivers waiting in the queue or at the loading gate are drawing a salary but are productively idle and unable to generate revenue during that time. What would be the hourly cost savings to the firm associated with employing two loaders instead of one?

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