Quantitative Methods: Probability questions

I have 5 probability problems that I need to be added to a template

MAT540Homework

Week 3

Page 1 of 3

MAT540

Week 3 Homework

Chapter 14

1. The Hoylake Rescue Squad receives an emergency call every 1, 2, 3, 4, 5, or 6 hours, according to

the following probability distribution. The squad is on duty 24 hours per day, 7 days per week:

Time Between

Emergency Calls (hr.)
Probability

1 0.15

2 0.10

3 0.20

4

0.25

5 0.20

6 0.10

1.00

a. Simulate the emergency calls for 3 days (note that this will require a “running” , or cumulative,

hourly clock), using the random number table.

b. Compute the average time between calls and compare this value with the expected value of the

time between calls from the probability distribution. Why are the result different?

2. The time between arrivals of cars at the Petroco Services Station is defined by the following

probability distribution:

Time Between
Emergency Calls (hr.)
Probability

1

0.35

2 0.25

3 0.20

4 0.20

1.00

MAT540 Homework
Week 3

Page 2 of 3

a. Simulate the arrival of cars at the service station for 20 arrivals and compute the average time

between arrivals.

b. Simulate the arrival of cars at the service station for 1 hour, using a different stream of random

numbers from those used in (a) and compute the average time between arrivals.

c. Compare the results obtained in (a) and (b).

3. The Dynaco Manufacturing Company produces a product in a process consisting of operations of

five machines. The probability distribution of the number of machines that will break down in a

week follows:

Machine Breakdowns

Per Week
Probability

0 0.10

1 0.20

2 0.15

3 0.30

4 0.15

5 0.10

1.00

a. Simulate the machine breakdowns per week for 20 weeks.

b. Compute the average number of machines that will break down per week.

4. Simulate the following decision situation for 20 weeks, and recommend the best decision.

A concessions manager at the Tech versus A&M football game must decide whether to have the

vendors sell sun visors or umbrellas. There is a 30% chance of rain, a 15% chance of overcast skies,

and a 55% chance of sunshine, according to the weather forecast in college junction, where the

game is to be held. The manager estimates that the following profits will result from each decision,

given each set of weather conditions:

MAT540 Homework
Week 3

Page 3 of 3

Decision Weather Conditions

Rain

0.35

Overcast

0.25

Sunshine

0.40

Sun visors $-400 $-200 $1,500

Umbrellas 2,100 0 -800

5. Every time a machine breaks down at the Dynaco Manufacturing Company (Problem 3), either 1, 2,

or 3 hours are required to fix it, according to the following probability distribution:

Repair Time (hr.) Probability

1 0.20

2 0.50

3 0.30
1.00

Simulate the repair time for 20 weeks and then compute the average weekly repair time.

2

>P

1

of

(lower bound)

Time between calls

Time between calls

.1

1 1

2 2

3

4

0.2 5 5
0.1

6

1

Time =

																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																							Hoylake Rescue Squad
	Probability				Time between calls
				Simulation
					P(x)							Cumulative	simulation Number								RN		Cumulative clock
																										0								5
		0.1
						0.2										3
			0.25									4
					6
				7
				8
EV =					9
	Average					10
				11
				12
				13
				14
				15
				16
				17
				18
				19
				20

P2

Simulation

Probability Cumulative

5

1

RN Time between calls RN Time between calls Cumulative clock

0.25 2 1
0.2 3 2
0.2 4 3
1 4
5

6

7

8

9

10

11
12
13
14
15
16
17
18
19
20

Petroco service
Time between arrival (min)
			0.3	Counts
a. Avg Arrival time
b. Avg. arrival time
Compare a. and b.

P3

Simulation

P(x) Cumulative

RN

0.1 0 1
0.2 1 2

2 3

0.3 3 4
0.15 4 5
0.1 5 6
1 7
8
9

10

11

12
13
14
15
16
17
18
19
20

	Dynaco Manufacturing
Probability breakdown per week
		Breakdown			Week	Breakdowns
			0.15
Simulated avg. breakdown
Average breakdowns =

P4

or

?

Simulation

P(x) Cumulative Sun Visor Week RN

RN

1

0.25

2

3

1 4
5

6
P(x) Cumulative Umbrella 7
0.35

8

0.25 0 9
0.4

10

1 11

12
13
14
15
16
17
18
19
20

Average

	Sun Visor	Umbrella
SunVisor ($)	Umbrella ($)
	0.35	-400
-200
	0.4	1500
2100
-800

P5

Dynaco Manufacturing

Simulation Breakdown

P(x) Cumulative

Week RN

RN

P(x) Cumulative Breakdown

0.2 1 1 0 0.1 0

2 2 0 0.2 1

0.3 3 3 0 0.15 2
1 4 0 0.3 3
5 0 0.15 4
6 0 0.1 5

7 0

8 0
9 0

10 0

11 0
12 0
13 0
14 0
15 0
16 0
17 0
18 0
19 0
20 0

Table from P3
Repair Time
Repair (hrs)	Breakdown #	Repair time/breakdown	Repair Time/week
0.5
Simulated avg. repair time
Theoretically calculated
Average repair time