1. Use the following table to calculate the expected value. The following table describes the possible outcomes and their associated probabilities.
|
x |
P(x) |
|
($20) |
0.05 |
|
($5) |
0.1 |
|
$0 |
0.15 |
|
$10 |
0.35 |
|
$15 |
0.2 |
|
$30 |
0.15 |
The first 2 outcomes are losses (negative values).
What is the expected value of this probability distribution?
(1 point)
2. In a certain school (enrollment in the school is well over 1000 students), 40% are male. If you randomly select 8 students:
A. What is the probability exactly 3 will be male?
B. What is the probability that at least 2 will be male?
(1 point each = 2 points)
3. In a certain hospital ER the average amount of patients that enter in a 15 minute interval is 3.6. If each 15 minute interval is independent of one another, and the amount of patients that show up in one 15 minute interval does not influence or change the number of patients that show up in any other 15 minute interval. Arrivals occur smoothly throughout each 15 minute interval. Answer the following questions:
A. What is the probability that at most 2 patients will show up in a 15 minute interval?
B. What would be your new mean, if we were to use this information to figure out probabilities during a 5 minute interval?
C. Using the new mean in part B, what is the probability that more than 1 patient will show up in a 5 minute interval?
In: Statistics and Probability
Submit your calculation/answer for Part 1 and MATLAB code/result/answer for Part 2 below:
Servicing Customers A supermarket you work part-time at has one express lane open from 5 to 6 PM on weekdays (Monday through Friday). This time of the day is usually the busiest since people tend to stop on their way home from work to buy groceries. The number of items allowed in the express lane is limited to 10 so that the average time to process an order is fairly constant at about 1 minute. The manager of the supermarket notices that there is frequently a long line of people waiting and hears customers grumbling about the wait. To improve the situation he decides to open additional express lanes during this time period. If he does, however, he will have to "pull" workers from other jobs around the store to serve as cashiers. Hence, he is reluctant to open more lanes than necessary. Knowing that you are a college student studying probability, your manager asks you to help him decide how many express lanes to open. His requirement is that there should be no more than one person waiting in line 95% of the time. With the task at hand, you set out to study the problem first. You start by counting the number of customer arrival in the express lane on a Monday from 5 to 6pm. There are a total of 81 arrivals. You repeat the experiment on the following four days (Tuesday through Friday) and note the total arrivals of 68, 72, 61 and 66 customers, respectively.
Part 1: Analysis (2% of final grade) In order to solve the problem, you decide to answer the following set of questions:
1)
2)Ans= 1.16
3) Ans = 67.71%
4) Ans= 96.53%
Part 2: Simulation (2% of final grade)
Before telling your manager your recommendation, you decide to simulate the problem first to verify your solution:
1) You decide to approximate the customer arrival process as follows. You treat each one-second interval as a Bernoulli trial. Assign it to be a one, if there is a customer arrives during that interval, zero if no customer arrives.
2) You count the number of customers arrives during a one-minute interval.
3) You count the total number of minutes out of a one-hour period that have two or fewer customers arrive. Does this number give your probability close to your calculation in Part 1 Prob 3?
4) Now based on your answer to Part 1 Prob 4, assign the arrivals in Part 2 Prob 1 with equal probabilities to the number of express lanes you recommend.
5) You count the number of customers arrives at each lane during a one-minute interval.
6) You count the total number of minutes out of a one-hour period that all lanes have two or fewer customers arrive. Does this number give you probability close to your calculation in Part 1 Prob 4?
Done in MatLab please.
In: Electrical Engineering
Suppose you are part of the analytics team for the online retailer Macha Bucks which sells two types of tea to its online visitors: Rouge Roma (RR) and Emerald Earl (EE). Everyday approximately 10,000 people visit the site over a 24 hour period. For simplicity suppose we consider the “buy one or don’t buy” (BODB) market segment of customers which when they visit the site will conduct one of the following actions: (a) buy one order of RR, (b) buy one order of EE, or (c) don’t buy (DB) anything. You have been tasked with determining customer behavior on the website for the BODB segment using a random sample of 35 visits.
In the dataset for the random sample, each row corresponds to a random visitor. For each visitor we provide both the visitor’s action as well as the profit earned on the transaction. In the action column:
if the visitor buys one order of RR, we see a RR,
if the visitor buys one order of EE, we see an EE,
if the visitor doesn’t buy anything, we see a DB.
Note that even if two customers buy the same product, the profit can differ due to the shipping costs, promotions, or coupons that are applied
Random Sample of Data
1=yes, 0 = no
Transaction ID
Action
Profit ($)
Bought RR?
Bought EE?
Didn't Buy?
Profit RR ($)
Profit EE ($)
1
RR
8.43
1
0
0
.
0.00
2
DB
0.00
0
0
1
0.00
0.00
3
EE
1.75
0
1
0
0.00
1.75
4
DB
0.00
0
0
1
0.00
0.00
5
EE
4.37
0
1
0
0.00
4.37
6
EE
5.79
0
1
0
0.00
5.79
7
RR
6.27
1
0
0
6.27
0.00
8
RR
6.22
1
0
0
6.22
0.00
9
DB
0.00
0
0
1
0.00
0.00
10
EE
4.49
0
1
0
0.00
4.49
11
RR
10.54
1
0
0
10.54
0.00
12
EE
3.79
0
1
0
0.00
3.79
13
DB
0.00
0
0
1
0.00
0.00
14
DB
0.00
0
0
1
0.00
0.00
15
RR
9.03
1
0
0
9.03
0.00
16
EE
3.54
0
1
0
0.00
3.54
17
DB
0.00
0
0
1
0.00
0.00
18
DB
0.00
0
0
1
0.00
0.00
19
EE
5.02
0
1
0
0.00
5.02
20
DB
0.00
0
0
1
0.00
0.00
21
EE
3.60
0
1
0
0.00
3.60
22
DB
0.00
0
0
1
0.00
0.00
23
EE
2.61
0
1
0
0.00
2.61
24
RR
11.75
1
0
0
11.75
0.00
25
RR
12.22
1
0
0
12.22
0.00
26
DB
0.00
0
0
1
0.00
0.00
27
DB
0.00
0
0
1
0.00
0.00
28
EE
6.17
0
1
0
0.00
6.17
29
RR
8.83
1
0
0
8.83
0.00
30
DB
0.00
0
0
1
0.00
0.00
31
DB
0.00
0
0
1
0.00
0.00
32
DB
0.00
0
0
1
0.00
0.00
33
DB
0.00
0
0
1
0.00
0.00
34
RR
14.16
1
0
0
14.16
0.00
35
EE
6.06
0
1
0
0.00
6.06
PARTS
a) What could be an appropriate probability distribution to use for
modeling the number of visitors that the website has in an
hour?
b) What parameters would you use for the probability
distribution?
c) Using that distribution, determine the probability that more
than 600 people visit the site in an hour.
a) What could be an appropriate probability distribution to use for
modeling the number of seconds between customer visits?
b) What parameters would you use for the probability
distribution?
c) Using that distribution, determine the probability that the time
between customer visits to the website is less than 10 seconds.
a) What could be an appropriate probability distribution to use for
modeling the number of website visitors from 100 visitors that do
not buy anything?
b) What parameters would you use for the probability
distribution?
c) Using that distribution, determine the probability that from
among 100 customers, it turns out that 30 or more customers do not
buy anything.
d) What is the average number of visitors (from among 100
customers) that do not buy anything?
e) What is the standard deviation of the number of visitors (from among 100 customers) that do not buy anything?
What is the average profit from among 100 random customers that
visit the site?
Please explain your answer or show your calculations.
In: Statistics and Probability
Suppose you are part of the analytics team for the online retailer Macha Bucks which sells two types of tea to its online visitors: Rouge Roma (RR) and Emerald Earl (EE). Everyday approximately 10,000 people visit the site over a 24 hour period. For simplicity suppose we consider the “buy one or don’t buy” (BODB) market segment of customers which when they visit the site will conduct one of the following actions: (a) buy one order of RR, (b) buy one order of EE, or (c) don’t buy (DB) anything. You have been tasked with determining customer behavior on the website for the BODB segment using a random sample of 35 visits.
In the dataset for the random sample, each row corresponds to a random visitor. For each visitor we provide both the visitor’s action as well as the profit earned on the transaction. In the action column:
if the visitor buys one order of RR, we see a RR,
if the visitor buys one order of EE, we see an EE,
if the visitor doesn’t buy anything, we see a DB.
Note that even if two customers buy the same product, the profit can differ due to the shipping costs, promotions, or coupons that are applied
Random Sample of Data
1=yes, 0 = no
Transaction ID
Action
Profit ($)
Bought RR?
Bought EE?
Didn't Buy?
Profit RR ($)
Profit EE ($)
1
RR
8.43
1
0
0
.
0.00
2
DB
0.00
0
0
1
0.00
0.00
3
EE
1.75
0
1
0
0.00
1.75
4
DB
0.00
0
0
1
0.00
0.00
5
EE
4.37
0
1
0
0.00
4.37
6
EE
5.79
0
1
0
0.00
5.79
7
RR
6.27
1
0
0
6.27
0.00
8
RR
6.22
1
0
0
6.22
0.00
9
DB
0.00
0
0
1
0.00
0.00
10
EE
4.49
0
1
0
0.00
4.49
11
RR
10.54
1
0
0
10.54
0.00
12
EE
3.79
0
1
0
0.00
3.79
13
DB
0.00
0
0
1
0.00
0.00
14
DB
0.00
0
0
1
0.00
0.00
15
RR
9.03
1
0
0
9.03
0.00
16
EE
3.54
0
1
0
0.00
3.54
17
DB
0.00
0
0
1
0.00
0.00
18
DB
0.00
0
0
1
0.00
0.00
19
EE
5.02
0
1
0
0.00
5.02
20
DB
0.00
0
0
1
0.00
0.00
21
EE
3.60
0
1
0
0.00
3.60
22
DB
0.00
0
0
1
0.00
0.00
23
EE
2.61
0
1
0
0.00
2.61
24
RR
11.75
1
0
0
11.75
0.00
25
RR
12.22
1
0
0
12.22
0.00
26
DB
0.00
0
0
1
0.00
0.00
27
DB
0.00
0
0
1
0.00
0.00
28
EE
6.17
0
1
0
0.00
6.17
29
RR
8.83
1
0
0
8.83
0.00
30
DB
0.00
0
0
1
0.00
0.00
31
DB
0.00
0
0
1
0.00
0.00
32
DB
0.00
0
0
1
0.00
0.00
33
DB
0.00
0
0
1
0.00
0.00
34
RR
14.16
1
0
0
14.16
0.00
35
EE
6.06
0
1
0
0.00
6.06
PARTS
a) What could be an appropriate probability distribution to use for
modeling the number of visitors that the website has in an
hour?
b) What parameters would you use for the probability
distribution?
c) Using that distribution, determine the probability that more
than 600 people visit the site in an hour.
a) What could be an appropriate probability distribution to use for
modeling the number of seconds between customer visits?
b) What parameters would you use for the probability
distribution?
c) Using that distribution, determine the probability that the time
between customer visits to the website is less than 10 seconds.
a) What could be an appropriate probability distribution to use for
modeling the number of website visitors from 100 visitors that do
not buy anything?
b) What parameters would you use for the probability
distribution?
c) Using that distribution, determine the probability that from
among 100 customers, it turns out that 30 or more customers do not
buy anything.
d) What is the average number of visitors (from among 100
customers) that do not buy anything?
e) What is the standard deviation of the number of visitors (from among 100 customers) that do not buy anything?
What is the average profit from among 100 random customers that
visit the site?
Please explain your answer or show your calculations.
In: Statistics and Probability
1. A soccer player will kick a ball 80 times during practice.
Assume that the kicks are independent of each
other, and the probability that he scores is 0.6 (60% chance that
the ball goes into the goalpost and 40%
chance that the ball does not go into the goalpost).
Let X be the number of successful goals (number of scores) out of
the 80 kicks.
(a) What is the distribution of X?
(b) Write the pmf f(x) and name its parameters.
(c) What key assumption of the kicks is needed to determine this
distribution?
(d) What is the expected number of kicks that go into the goalpost?
Interpret this value for the soccer
player (in a sentence or two).
(e) What is the expected number of kicks that do not go into the
goal post? Interpret this value for
the soccer player (in a sentence or two).
(f) Say each kick is blocked by the opponent goal keeper 30% of the
time regardless of whether the ball
was going in or out of the goalpost. What is the expected number of
blocks? What is the variance
of the number of blocks?
(g) Now say each kick that was supposed to go into the goal post is
rebounded by another player 50%
of the time and each kick that was not going into the goalpost is
rebounded by another player 10%
of the time. What is the expected number of rebounds?
In: Statistics and Probability
You will develop code in the StickWaterFireGame class only. Your tasks are indicated by a TODO. You will also find more instructions in the form of comments that detail more specifically
what each part of the class is supposed to do, so make sure to
read them. Do not delete or modify the method headers in the
starter code!
Here is an overview of the TODOs in the class.
TODO 1: Declare the instance variables of the class. Instance variables are private variables that keep the state of the game. The recommended instance variables are:
1. A variable, “rand” that is the instance of the Random class.
It will be initialized in a constructor (either seeded or unseeded)
and will be used by the getRandomChoice() method.
2. A variable to keep the player’s cumulative score. This is the
count of the number of times the player (the user) has won a round.
Initial value: 0.
3. A variable to keep the computer’s cumulative score. This is the count of the number of times the computer has won a round. Initial value: 0.
4. A variable to keep the count of the number of rounds that have been played. Initial value: 0.
5. A variable to keep track of if the player has won the current round (round-specific): true or false. This would be determined by the playRound method. Initial value: false.
6. A variable to keep track of if the player and computer have tied in the current round (round-specific): true or false. This would be determined by the playRound method. Initial value: false.
The list above contains the minimum variables to make a working program. You may declare more instance variables as you wish.
TODOs 2 and 3: Implement the constructors. The constructors assign the instance variables to their initial values. In the case of the Random class instance variable, it is initialized to a new instance of the Random class in the constructors. If a constructor has a seed parameter, the seed is passed to the Random constructor. If the constructor does not have a seed parameter, the default (no parameter) Random constructor is called.
TODO 4: This method returns true if the inputStr passed in is one of the following: "S", "s", "W", "w", "F", "f", false otherwise. Note that the input can be upper or lower case.
TODOs 5, 6, 7, 8, 9, 10 These methods just return the values of their respective instance variables.
TODO 11: This is a private method that is called by the playRound method. It uses the instance variable of the Random class to generate an integer that can be “mapped” to one of the three Strings: "S", "W", "F", and then returns that String, which represents the computer’s choice.
TODO 12: The playRound method carries out a single round of play of the SWF game. This is the major, “high-level” method in the class. This method does many tasks including the following steps:
1. Reset the variables that keep round-specific data to their
initial values.
2. Assign the computer’s choice to be the output of the
getRandomChoice method.
3. Check that the player’s choice is valid by calling isValidInput.
If the player's input is not valid, the computer wins by default.
This counts as a round, so the number of rounds is incremented. 4.
If the player’s choice is valid, the computer's choice is compared
to the player's choice in a series of conditional statements to
determine if there is a tie, or who wins this round of play
according to the rules of the game:
S beats W
W beats F
F beats S
Both have the same choice- a tie.
The player and computer scores are updated depending on who wins. In the event of a tie, neither player has their score updated. Finally, the number of rounds of play is incremented.
Note: Do not duplicate the isValidInput or the getRandomChoice code in playRound. Call these methods from playRound. The point is that delegating tasks to other methods makes the code easier to read, modify and debug.
THE CODE:
1 import java.util.Random;
2
3 /* This class ecapsulates the state and logic required to play
the
4 Stick, Water, Fire game. The game is played between a user and
the computer.
5 A user enters their choice, either S for stick, F for fire, W for
water, and
6 the computer generates one of these choices at random- all
equally likely.
7 The two choices are evaluated according to the rules of the game
and the winner
8 is declared.
9
10 Rules of the game:
11 S beats W
12 W beats F
13 F beats S
14 no winner on a tie.
15
16 Each round is executed by the playRound method. In addition to
generating the computer
17 choice and evaluating the two choices, this class also keeps
track of the user and computer
18 scores, the number of wins, and the total number of rounds that
have been played. In the case
19 of a tie, neither score is updated, but the number of rounds is
incremented.
20
21 NOTE: Do not modify any of the code that is provided in the
starter project. Additional instance variables and methods
22 are not required to make the program work correctly, but you may
add them if you wish as long as
23 you fulfill the project requirements.
24
25 */
26 public class StickWaterFireGame {
27
28 // TODO 1: Declare private instance variables here:
29
30
31 /* This constructor assigns the member Random variable, rand,
to
32 * a new, unseeded Random object.
33 * It also initializes the instance variables to their default
values:
34 * rounds, player and computer scores will be 0, the playerWins
and isTie
35 * variables should be set to false.
36 */
37 public StickWaterFireGame() {
38 // TODO 2: Implement this method.
39
40 }
41
42 /* This constructor assigns the member Random variable, rand,
to
43 * a new Random object using the seed passed in.
44 * It also initializes the instance variables to their default
values:
45 * rounds, player and computer scores will be 0, the playerWins
and isTie
46 * variables should be set to false.
47 */
48 public StickWaterFireGame(int seed) {
49 // TODO 3: Implement this method.
50
51 }
52
53 /* This method returns true if the inputStr passed in is
54 * either "S", "W", or "F", false otherwise.
55 * Note that the input can be upper or lower case.
56 */
57 public boolean isValidInput(String inputStr) {
58 // TODO 4: Implement this method.
59 return false;
60 }
61
62 /* This method carries out a single round of play of the SWF
game.
63 * It calls the isValidInput method and the getRandomChoice
method.
64 * It implements the rules of the game and updates the instance
variables
65 * according to those rules.
66 */
67 public void playRound(String playerChoice) {
68 // TODO 12: Implement this method.
69 }
70
71 // Returns the choice of the computer for the most recent round
of play
72 public String getComputerChoice(){
73 // TODO 5: Implement this method.
74 return null;
75 }
76
77 // Returns true if the player has won the last round, false
otherwise.
78 public boolean playerWins(){
79 // TODO 6: Implement this method.
80 return false;
81 }
82
83 // Returns the player's cumulative score.
84 public int getPlayerScore(){
85 // TODO 7: Implement this method.
86 return 0;
87 }
88
89 // Returns the computer's cumulative score.
90 public int getComputerScore(){
91 // TODO 8: Implement this method.
92 return 0;
93 }
94
95 // Returns the total nuber of rounds played.
96 public int getNumRounds(){
97 // TODO 9: Implement this method.
98 return 0;
99 }
100
101 // Returns true if the player and computer have the same score
on the last round, false otherwise.
102 public boolean isTie(){
103 // TODO 10: Implement this method.
104 return false;
105 }
106
107 /* This "helper" method uses the instance variable of Random to
generate an integer
108 * which it then maps to a String: "S", "W", "F", which is
returned.
109 * This method is called by the playRound method.
110 */
111 private String getRandomChoice() {
112 // TODO 11: Implement this method.
113 return null;
114 }
115 }
116
In: Computer Science
An individual faces two alternatives for an
investment: Asset A has the following probability return
schedule:
| Probability of return | Return (yield) % |
| 0.20 | 10 |
| 0.30 | 8 |
| 0.10 | - 4 |
| 0.40 | - 1 |
Asset B with a certain return of 2.0%. Calculate the expected
return on Asset A.Would a risk averse investor ever choose
investment A over investment B? Why or why not?
[Hint: You need to calculate and compare expected values to
successfully answer this question!]
In: Finance
Quantitative Problem: You are given the following probability distribution for CHC Enterprises:
| State of Economy | Probability | Rate of return | |
| Strong | 0.15 | 20 | % |
| Normal | 0.55 | 9 | % |
| Weak | 0.30 | -5 | % |
What is the stock's expected return? Do not round intermediate calculations. Round your answer to two decimal places.
%
What is the stock's standard deviation? Do not round intermediate calculations. Round your answer to two decimal places.
%
What is the stock's coefficient of variation? Do not round intermediate calculations. Round your answer to two decimal places.
In: Finance
Monty Hall Problem: How was probability used in arriving at the answer? What probability ideas does this demonstrate and use? Explain and give examples. You may use other sources as well but make sure to cite them.
In: Statistics and Probability
Assume that the following probability distribution exists for automobile damages
| Possible Outcomes for Damages | Probability |
| $0 | 50% |
| 600 | 30% |
| 2,000 | 10% |
| 7,000 | 6% |
| 11,000 | 4% |
What is the expected value for damages?
A. $12.40
B. $124
C. 1,240
D. 12,400
Can someone please explain how you got the answer. I'm stuck
In: Finance