Problem 11-30 (Algorithmic)

A large insurance company maintains a central computing system that contains a variety of information about customer accounts. Insurance agents in a six-state area use telephone lines to access the customer information database. Currently, the company's central computer system allows three users to access the central computer simultaneously. Agents who attempt to use the system when it is full are denied access; no waiting is allowed. Management realizes that with its expanding business, more requests will be made to the central information system. Being denied access to the system is inefficient as well as annoying for agents. Access requests follow a Poisson probability distribution, with a mean of 34 calls per hour. The service rate per line is 12 calls per hour.

1. What is the probability that 0, 1, 2, and 3 access lines will be in use? Round your answers to 4 decimal places.

   j	Pj
   0
   1
   2
   3

2. What is the probability that an agent will be denied access to the system? Round your answers to 4 decimal places.
   
   
3. What is the average number of access lines in use? Round your answers to 4 decimal places.
   
   L =
4. In planning for the future, management wants to be able to handle λ = 42 calls per hour; in addition, the probability that an agent will be denied access to the system should be no greater than the value computed in part (b). How many access lines should this system have? (already calculated 5 to be the answer)

This is all one question.

State the number of friends (or connections) that you have on Facebook (or Linkedin). In case you have more than 365 friends or connections, think of an alternative, smaller group of friends or relatives. What is the chance that there are at least 2 people among your friends (or connections) with the same birthday (same day, not same year)? Let's find out. Please respond with an estimate of the probability that this will happen. This estimate can be intuitive or you can do some calculations: both ways are ok. In case you know the birthdays of your friends (or connections), check if your estimate of the probability corresponds to the reality.

A street performer approaches you to make a bet. He shows you three cards: one that is blue on both sides, one that is orange on both sides, and one that is blue on one side and orange on the other. He puts the cards in the bag, pulls out one, and puts it on the table. Both of you can see that the card is blue on top, but haven't seen the other side. The street performer bets you $50 that the other side of the card is also blue. Should you take the bet and WHY?

Now that the previous two questions have gotten you thinking about probability, how does probability apply to your (desired) profession?

All potential criminals are alike. Each has a benefit B of committing a crime, where B = $10,000. The cost, to the criminal, of being punished is T =$1,000 for each year spent in prison. The probability of a criminal being caught and punished is p. Let S represent the number of years spent in prison (i.e., the sentence). Suppose there are 100 potential criminals. Each chooses whether to commit this crime which has a social harm cost of $100,000. Suppose criminals are caught with a 15% probability. The cost of prison is $5,000 per prisoner, per year.

(a) Write down the condition for a rational criminal to commit a crime. What is the optimal choice of sentence, S? What is the total social cost associated with this choice? [5 marks]

(b) Now suppose that there are an additional 50 criminals who are irrational and therefore, always commit crimes. How does your answer to part (a) change? [5 marks]

(c) Now suppose you can choose from one of the following two police forces:(i) Catches criminals with probability 0.25 and costs $500,000.(ii) Catches criminals with probability 0.15 and costs $250,000.The social cost of crime is: prison costs plus social harm plus police costs less the benefit to the criminal. In a world with 100 rational criminals and 50 irrational criminals, which of these police forces would you choose, and what level of sentence S would you choose. Justify for your answer. [10 marks]

1. Let’s use Excel to simulate rolling two dice and finding the rolled sum.

• Open a new Excel document.

• Click on cell A1, then click on the function icon fx and select Math&Trig, then select RANDBETWEEN.

• In the dialog box, enter 1 for bottom and enter 6 for top.

• After getting the random number in the first cell, click and hold down the mouse button to drag the lower right corner of this first cell, and pull it down the column until 25 cells are highlighted. When you release the mouse button, all 25 random numbers should be present.

• Repeat these four steps for the second column, starting in cell B1.

• Put the rolled sum of two dice in the third column: Highlight the first two cells in the first row and click on AutoSum icon. Once you receive the sum of two values in the third cell, drag the lower right corner of this cell, C1, down to C25. This will copy the formula for all 25 rows. We now have 25 trials of our experiment.

• Once these steps are completed, attach a screenshot of your Excel file to your assignment.

(a) Find the probability that the rolled sum of both dice is 5.

(b) Based on the results of our experiment of 25 trials, obtain the relative frequency approximation to the probability found in (a).

(c) Repeat the simulation for 50 and 100 trials, and calculate the relative frequency approximation to the probability in (a) for each. Which approximation has the closest value to the probability?

(d) Briefly explain how these experiments demonstrate the Law of Large Numbers.

Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 miligrams per deciliter (mg/dl) one hour after having a sugary drink. Sheila's measured glucose level one hour after the sugary drink varies according to the Normal distribution with μμ = 125 mg/dl and σσ = 15 mg/dl.

(a) If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes?
(b) If measurements are made on 7 separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Sheila is diagnosed as having gestational diabetes?

Andrew plans to retire in 36 years. He plans to invest part of his retirement funds in stocks, so he seeks out information on past returns. He learns that over the entire 20th century, the real (that is, adjusted for inflation) annual returns on U.S. common stocks had mean 8.7% and standard deviation 20.2%. The distribution of annual returns on common stocks is roughly symmetric, so the mean return over even a moderate number of years is close to Normal.

(a) What is the probability (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 36 years will exceed 11%?
(b) What is the probability that the mean return will be less than 4%?

One state lottery game has contestants select 5 different numbers from 1 to 45. The prize, if all numbers are matched is 2 million dollars. The tickets are $2 each.

1) How many different ticket possibilities are there?

2) If a person purchases one ticket, what is the probability of winning? What is the probability of losing?

3) Occasionally, you will hear of a group of people going in together to purchase a large amount of tickets. Suppose a group of 30 purchases 6,000 tickets. a) How much would each person have to contribute? b) What is the probability of the group winning? Losing?

4) How much would it cost to “buy the lottery”, that is, buy a ticket to cover every possibility? Is it worth it?

5) Create a probability distribution table for the random variable x = the amount won/lost when purchasing one ticket.

6) In fair games, the expected value will be $0. This means that if the game is played many…many times, then one is expected to break even eventually. This is never true for Casino and Lottery games. Find the expected value of x = the amount won/lost when purchasing one ticket.

7) Interpret the expected value. See section 4.2 in the textbook for an example on how to interpret the expected value.

8) Fill in the following table using the expected value.

Please answer all questions! I will rate you!

At the nursery, there were 6 potted plants that were supposed to grow red flowers and 8 potted plants that were supposed to grow green flowers. Freddie bought 4 flower pots but cannot for the life of him remember what colour they were supposed to be. He decided that when the plants bloom, for every red flower that grows, he will buy a scratch ticket. Assume his scratch tickets have a 25% chance of being winners and the chance of winning is independent. Let X be the number of red flowers that bloom and Y be the number of winning scratch tickets.

1. (A) [1 mark]
   Identify the distribution of X.

2. (B) [3 marks]
   Find the joint probability distribution of X and Y .

3. (C) [2 marks]
   Find P (X = Y ).

There are only 8 software firms that design a certain kind of software. Of​ these, 3 use Protocol A in their software. A working group of 5 designers​ (no more than one from a​ firm, some firms were not​ represented) was formed to reach an a new protocol. Assume that the working group is chosen without regard to whether a​ designer's firm uses Protocol A.

a) What is the probability that exactly 0 of the designers on the working group will be from firms that use Protocol​ A? ​ (Use four decimal​ places.) 

b)What is the mean number of designers on the working group that are from firms that use Protocol​ A? ​ (Use one decimal place in your​ answer.)

c)What is the variance of the number of designers on the working group that are from firms that use Protocol​ A? ​ (Use two decimal places in your​ answer.)  

Do you tailgate the car in front of you? About 35% of all drivers will tailgate before passing, thinking they can make the car in front of them go faster. Suppose that you are driving a considerable distance on a two-lane highway and are passed by 8 vehicles. (a) Let r be the number of vehicles that tailgate before passing. Make a histogram showing the probability distribution of r for r = 0 through r = 8. Maple Generated Plot Maple Generated Plot Maple Generated Plot Maple Generated Plot

(b) Compute the expected number of vehicles out of 8 that will tailgate. (Round your answer to two decimal places.) . vehicles

(c) Compute the standard deviation of this distribution. (Round your answer to two decimal places.) vehicles

1) You own 6 songs by Adele, 4 by Katy Perry, and 5 by Lady Gaga. How many different playlists can you make that consist of 4 Adele songs, 3 Perry songs, and 2 Gaga songs, if you do allow repeated songs?

2) Kevin and Walter can't stop discussing their Probability course at every opportunity, which has been very disruptive of the regular discussions at the CCIS Teaching Seminar. Martin, who organizes the seminar, must now calculate the number of ways in which he can arrange the 10 participants in a line so that Kevin and Walter aren't sitting next to each other. What is that number?

3) In a race with 6 runners, how many possible finishing orders are there?