* On the average, 2 *cars will arrive at DRMC in *an hour*.

Compute the probability that *exactly 5 *cars will arrive in the next hour.

Compute the probability that *no more than 5 *cars will arrive in the next hour?

2. Miss Helen claims she has been unfairly dismissed by her employers. She consults a law firm owned by Mr. Habtamu, who agrees to take up her case. The law consulting firm, Owned by Mr. Habtamu advises her that if she wins her case she can expect a compensation of 90, 000 birr but if she loses she will receive nothing. The law consultant firm estimates his fee to be 20, 000 birr which she will have to pay whether she wins or loses. Under the rules of the relevant court she cannot be asked to pay her employer’s costs. As an alternative the consultant offer her a ‘no win no fee’ deal under which she pays no fee but if she wins her case, the consulting firm will take one-third of the compensation she receives. She can decide against bringing the case, which will incur no cost and result in no compensation. Based on the information given above:

Develop the payoff table to the above case

Advise Helen what to do:

Using the maximax decision rule

Using the maximin decision

Using the Minmax regret decision

Using the equal likelihood decision

3. During the registration period, St. Mary’s Univeristy has a technician in its service center to answer students’ The number of telephone calls arriving at this center follows a Poisson distribution with an approximate average rate of 10/h. The time required to answer one call follows an exponential distribution with an average of 4 min. Answer the following questions:

What is the average time between incoming calls?

What is the average number of calls that the technician can attend in 1h?

What is the probability of there being exactly four calls on hold at a given time?

What is the probability of the number of calls in the system exceeding 10?

4. Two companies A and B are competing for the same product. Their different strategies are given in the following pay-off matrix:

A

B

I

II

III

I

-2

14

-2

II

-5

-6

-4

III

-6

20

-8

Based on the information given above answer the following questions

Is there a saddle point in this game?

What are the optimal strategies for both player A and B?

What is the value of the game?

5. The average salary for DRMC lecturers is $30,000 per If the distribution is approximately normal with a standard deviation of $5000, what is the probability that the annual salary of a randomly selected lecturer from the college is

between $27,500 and $32,500 per year

less than $25,000 per year.

6. Consider the decision analysis problem having the following payoff table

State of

Nature

Probability

0.2

0.3

0.5

S1

S2

S3

A1

-100

10

100

A2

-10

20

50

A3

10

10

60

Which alternative would be chosen for each of the following decision criterion?

Maximax

Maximin

Minimax regret

Hurwicz (Realism) (α=0.4)

Expected monetary value

Expected opportunity loss

7. State University is about to play Ivy College for the state tennis The State team has two players (A and B), and the Ivy team has three players (X, Y, and Z). The following facts are known about the players’ relative abilities: X will always beat B; Y will always beat A; A will always beat Z. In any other match, each player has a ½ chance of winning. Before State plays Ivy, the State coach must determine who will play first singles and who will play second singles. The Ivy coach (after choosing which two players will play singles) must also determine who will play first singles and second singles. Assume that each coach wants to maximize the expected number of singles matches won by the team. Use game theory to determine optimal strategies for each coach and the value of the game to each team.

8. The New York City Council is ready to vote on two bills that authorize the construction of new roads in Manhattan and Brooklyn. If the two boroughs join forces, they can pass both bills, but neither borough by itself has enough power to pass a bill. If a bill is passed, then it will cost the taxpayers of each borough $1 million, but if roads are built in a borough, the benefits to the borough are estimated to be $10 The council votes on both bills simultaneously, and each councilperson must vote on the bills without knowing how anybody else will vote. Assuming that each borough supports its own bill, determine whether this game has any equilibrium points. Is this game analogous to the Prisoner’s Dilemma? Explain why or why not.

9. A bank has two tellers working on saving accounts. The first teller handles withdrawals only. The second teller handles deposits only, it has been found the service time distribution for the deposits and withdrawals both are exponential with mean service time 3 minutes per Depositors are found to arrive in a Poisson distribution throughout the day with a mean arrival rate of 16 per hour. Withdrawals also arrive in a Poisson fashion with a mean arrival rate of 14 per hour.

What would be the effect on the average waiting time for depositors and withdrawers if each teller could handle both withdrawals and deposits?

What would be the effect if this could only be by increasing the service time to 3.5 minutes?

10. Consider a problem hierarchy with 4 criteria on which the three alternatives are to be judged. For pairwise comparison the first criteria will be compared against the remaining 3, the second criteria against the remaining 2, the third criteria against the remaining 1 as it is clearly indicated in the given below table.

Table: 1 Pairwise comparison Matrix for the criteria

Besides, the pairwise comparison matrices of each decision alternative (DA) i.e. B1, B2 and B3) with respect to each criterion are given in the table below:

Based on the above given information:

Develop normalized matrices and priority vectors accordingly

Which decision alternatives will be selected by you as a decision maker and why?

Is there an inconsistency problem? If yes why? If no why not?