probability

Part 1

‘AirCampania’ is a regional airline operating in Italy that has collected data for the percentage available seats occupied on

its flights for four quarters: January-March, April- June, July-September, and October-December for the past 5 years, as

follows:

Year Quarter Seat Occupancy (%)

2011 January-March 22

April-June 34

July-September 19

October-December 26

2012 January-March 18

April-June 40

July-September 27

October-December 31

2013 January-March 26

April-June 28

July-September 30

October-December 35

2014 January-March 21

April-June 36

July-September 26

October-December 30

2015 January-March 25

April-June 37

July-September 25

October-December 30

1. Use an appropriate moving average to identify the trend for seat occupancy. Plot the trend on a time-series graph.

[15 marks]

2. Use the additive model to calculate the seasonal components for seat occupancy.

[8 marks]

3. Using the multiplicative model, for the same data related to the seat occupancy, the

Mean Absolute Deviation of the errors was found equal to 5.00. Compare the results from your additive model with the ones

from the multiplicative model. Which one is the most appropriate? Justify your answer.

[12 marks]

D. Use the additive model to forecast the seat occupancy percentage in the year 2016. [15 marks]

Part 2

In all sections of this part you must show your calculations, stating the formulae and the values of the parameters you have

used.

1. A company is making the decision of whether to market two new products. A new product, called ‘Pippo’, has the following

profit projections and associated probabilities:

Pippo

Profit Probabilities

160 0.10

120 0.25

700 0.20

200 0.15

-300 0.20

-800 0.10

A. Use an appropriate approach to make the decision of whether to market the new product. [3 marks]

The second product, called ‘Paperino’, has the following profit projections and associated probabilities:

Paperino

Profit Probabilities

85 0.10

60 0.25

45 0.20

25 0.15

15 0.20

10 0.10

2.

B. Which product should the company prefer? Clearly support your claim with appropriate measures. [6 marks]

The Health and Safety Advisor found that 25% of work-related injuries, occurred on Fridays and that 18% occurred in the last

hour of a day’s shift. It was also found that 3% of work-related injuries occurred in the last hour of Friday’s shift.

A. Represent with a Venn diagram the events ‘work-related injury occurs on Friday’ and ‘work-related injury occurs in the

last hour of the day’s shift’.

[4 marks]

2. What is the probability that a work-related injury that occurs on Friday does not occur in the last hour of the day’s

shift? Show this probability on the Venn diagram previously drawn.

[4 marks]

3. Are the events ‘work-related injury occurs on Friday’ and ‘work-related injury occurs in the last hour of the day’s

shift’ statistically independent?

[4 marks]

3. The Portsmouth Accounting Company offers helps to its clients to decide their planning. A specialist financial

planner will take every client through a detailed process examining the objectives, including her tolerance and the capacity

for risk. The company receives a mean of 15 applications per week for this service. Each review requires one day of a

financial planner’s time to prepare a good investment plan. Requests received during any week are assigned to a financial

planner for completion during the following week. If the financial plan is not completed during the second week the customer

will cancel.

1. How many financial planners should be hired so that the company can claim that 90% of the plans will be completed

during the second week? Assume a five-day working week.

[8 marks]

2. Suppose that the company decided to hire one less financial planner than determined in part (A). What is the

probability that customers would cancel given this staffing level?

[8 marks]

4. The ‘SogniD’Oro’ hotel has a total of 300 identical rooms. It has adopted an overbooking policy in order to maximise

the usage of its available capacity.

Assume that each potential customer holding a room reservation, independently of other customers, cancels the reservation or

simply does not show up at the hotel on a given night with probability 0.06.

1. What is the probability that on a given night at least 10 customers do not show up? [6 marks]

2. Find the largest number of room reservations that this hotel can book and still be at least 95% sure that everyone

who shows up at the hotel will have a room on a given night. [7 marks]