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MBA course. The university knows that the probability of a student completing the course successfully is 80%. We take a random sample of 4, independent students.
a. What is the probability exactly 0 students complete the course?
b. What is the probability exactly 1 student complete the course?
c. What is the probability exactly 2 students complete the course?
d. What is the probability exactly 3 students complete the course?
e. What is the probability 3 or (all) 4 of the students complete the course?
f. What is the expected number of students (out of the 4 sampled) that complete the
course?
g. What is the standard deviation of the number of students that complete the course?

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2.(1) The claim by a weight loss Company is that on average, the client will lose 10 pounds over ...

first 2 weeks. 50 people who joined the programme are sampled, their weight loss is 9 pounds with a standard deviation of 2.8 pounds. Can we conclude at the .05 level that a person joining the programme will lose less than 10 pounds?
(2) The following is a random sample of 90-day futures prices in dollars for 1 troy oz. of silver from The Wall Street Journal issues in May and June of 1997: 4.74, 4.77, 4.87, 4.91, 4.83, 4.72, 4.92, 4.86, 4.97, 4.71, 4.90, 4.93, 4.75, 4.88, 4.79, 4.83, 4.89.
Required:
a. Calculate the mean
b. Median
c. Standard deviation of the 90-day future price of silver data
(3) A mining company needs to estimate the average amount of copper ore per ton mined. A random sample of 50 tons gives a sample mean of 146,75 pounds. The population standard deviation is assumed to be 35.2 pounds.
Required:
a. Give a 95% confidence interval for the average amount of copper in the population of tons mined.
b. Give a 90% confidence interval for the average amount of coper per ton
c. Give a 99% confidence interval for the average amount of coper per ton
(4) An e-commerce Website gets 2,385 visitors on a particular day. Among these, 1790 visitors explore the products by looking at more pages at the site. Among these 1790 visitors who explore the products, 387 make a purchase.
Required:
a. If a visitor chosen at random from all those who visited the site, what is the probability that the visitor explored the products
b. If a visitor is chosen at random from all those who visited the site, what is the probability that the visitor made a purchase.
c. If a visitor is chosen at random from all those who explored the products, what is the probability that the visitor made a purchase.
d. Which of the preceding three probabilities is relevant to the design of the home page that leads to product page.

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of the quiz is 15 minutes.
Topics for the quiz are:
Probability Revision, Dynamic Programming, Markov Process, Markov Chain Properties, Hidden Markov Model, Undirected Graphical Method.
I am trying to send all the study materials, but file size is large and it isn't allowing me to send. Please message me for further details regarding study materials so that I will share them with you and you can have a look.

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rd deviation of 3.2 hours. A sample of 35 high school seniors is selected. Find the following values, if possible. Please round your answers to four decimal places. If you should not find a probability, please write NA for your answer.
What is the probability that the sample mean studying time is between 4 and 5 hours?
What is the probability that the sample mean studying time is less than 6 hours?
What is the probability that the sample mean studying time is more than 8.5 hours?

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nd three of them are broken. Two glasses are chosen at random from the box. Calculate the probability that
(1) both glasses are broken,
(2) none of the glasses are broken
(3) both glasses are broken or one of the glasses is broken

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particular micro-organism. She finds that 62% of the
samples have at least one micro-organism present. Assuming the micro-
organisms are randomly occurring, with an average rate of occurrence, a) estimate k, the average number of micro-organisms per sample. b) what is the probability that in the combined sea water of three samples
(15ml) she finds more than 2 micro-organisms?

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ions, Cartesian coordinates, graphs,
kinematics.
Trigonometric functions
Exponential and logarithmic
functions
Limits and differentiation
Algebra, vector and matrices
Combinatorics and Probability
This task is divided in 3 stages.
each stage will have its own milestone and will be released upon completion and review.
Please just bid if you are available for all the tasks, check carefully the times.
Complete a questionary with the math questions, approx. 7 pages every answer should be complete and extensive and fully explained in order to be fully understood the process. Approx. time 2 to 2.5 hours.
Task 2 will take place on
Monday, September 6, 2021 at: 1:00pm / 13:00
Coordinated Universal Time (UTC)
This task will include a similar questionary with A B OR C Answers, approx time 1.5 hours to 2 hours, This questions will be provided one by one and requires that we connect by chat live.
Task 3 will take place on
Monday, September 6, 2021 at: 1:00pm / 13:00
Coordinated Universal Time (UTC) after task2
Upon completion of task 2 a document similar to task 1 will be provided in pdf, this one need to be completed within 4 hours after provided approx. 7 pages every answer should be complete and extensive and fully explained in order to be fully understood the process.
For now I will provide task 1 document.

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When selecting one card from a standard deck, what is the probability of selecting a Jack AND a Club?
An urn contains 10 red balls, 6 green balls, 15 orange balls, and 14 blue balls.
If one ball is randomly drawn from the urn, what are the odds against the ball being green?
The odds against an event are 7 to 1. Find the probability that the event will occur?
The probability of an event is 7/10. Find the odds against that event.

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9.Using a sample of SAT scores (n = 5000) with an average score of 600 and a standard deviation of ...

, determine the probability that a randomly selected score will be between a 450 and 700.

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thing else.
The probability a player makes a free throw in an away game is 50%, independently of everything else.
Half of the games are home games.
What is the probability that when the player throws two free throws she will make both?
Assuming the player makes the first free throw, what is the probability she will make the second? Are the two events independent?
Assuming that the player made both free throws, what is the probability it was a home game?

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of Sport-Equip Ltd, a company that sells sports equipment. Some of the users
who visit the website end up buying sports equipment from the website while others are
simply browsing in order to obtain product information.
(a) Clearly explain and justify which probability distribution you would use to
describe the number of Internet users who visit the website of Sport-Equip Ltd
in a one hour period. [There is no need to calculate any probabilities for this part
of the question]
(5 marks)
(b) What is the probability that during any half-hour period, there will be less than 3
visitors to the website?
(5 marks)
(c) What is the probability that during any two-hour period, there will be more than
15 visitors to the website?
(5 marks)
(d) If a user has just visited the website, find the probability that the website will
have another visitor within the next 10 minutes. In your answer, state the
probability distribution you have used and explain your choice.
(4 marks)
(e) It is estimated that 40% of Internet users who visit Sport-Equip Ltd’s website
buy a product from the company. If 100 users visit the website over a given
period of time, find the probability that more than 50 of them will buy a product
from the company. In your answer, state the probability distribution you have
used and explain your choice

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. If there is an intruder, system A sounds an alarm with probability 0.85 and system B sounds an alarm with probability 0.95. If there is no intruder, system A sounds an alarm with probability 0.20 and system B sounds an alarm with probability 0.15 . Now, assume that the probability of an intruder is 0.4 . If both systems sound an alarm, what is the probability that there is an intruder?

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13.There are 5 red balls, 3 blue balls and 2 green balls in a box. Suzie takes at random a ball ...

he box and gives it to her friend. She then takes at
random another ball from the box.
2.1.1 Write down the sample space of this experiment
qo
2.1.2 Work out the probability that both balls are of the same colour.

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14.In a box there are 4 red, 4 blue and 3 white marbles. Two marbles are selected at random and ...

or is noted. Find the probability that neither is white given that neither is red.Round your answer to 4 decimal places
A place kicker in the NFLmakes 78% of his field goals. If the outcomes are independent, what is the probability that he makes exactly 6 of the next 8 field goals? Round your answer to 4 decimal places
There are 5 white mice and 3 gray mice in a cage. Three mice are selected at random and their colors are noted. Find the expected number of white mice.Round your answer to 2 decimal places

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ty distribution:
x 3 5 8 10
pX(x) k k k 2k
i. Determine the constant k and hence write down the probability
distribution of X.
ii. Find E(X), the expected value of X.
iii. Find Var(X), the variance of X.

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16.A bag of 11 marbles contains 7 marbles with red on them, 3 with blue on them, 5 with green ...

and 4 with red and green on them. What is the probability that a randomly chosen marble has either green or red on it? Note that these events are not mutually exclusive. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

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y aces or twos, you lose the game immediately. You also lose if you draw picture cards(J,Q,K) more than twice. In this question, you’ll study the probability of winning this game.(a) What is the probability of drawing no aces or twos after thirteen draws?(b) Given you have drawn thirteen times, none of which is aces or twos, what is the probability that you draw at most two picture cards?(c) What is the probability to win this game?
12. Suppose you are tossing an unbiased coin for100times.(a) What is the probability of getting50heads and50tails?(b) LetXbe the random variable counting the number of heads you observe in this exper-iment. What is the expected value ofX? What is the variance ofX? What is thestandard deviation ofX?
13. The following are probability distributions for two random variablesX,Y.
kPr(X=k)
0,0.4
1,0.3
2,0.3
kPr(Y=k)
0,0.5
1,0.3
2,0.2
(a) Construct the probability distribution table for the random variableXY.(b) Find E[X],E[Y] and E[XY]. Is is true that E[XY] =E[X]E[Y]?(c) Find the variances σ2X,σ2Y,σ2XY of X,Y and XY. Is it true that σ2XY=σ2Xσ2Y?
14. The aliens who are fond of gambling came back to play another game with you. In this game, you first toss a coin5times. If you observe3or fewer tails, you roll a die3times. If youobserve4or more tails, you roll a die20times. What is the probability that you end up with at most two6’s in your dice rolls?
15. (Challenge question, worth2points) You have two bags, each of which contains10marbles.Each time you remove a marble from a random bag. What is the probability that after one of the bags is emptied, there are still exactly3marbles in the other bag?

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18.There are 4 sets of balls numbered 1 through 12 placed in a bowl. If 4 balls are randomly chosen ...

hout replacement, find the probability that the balls have the same number. Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

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than 7.33 is 0.6738, then what is the mean of x?

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- 1); I{x>1}, theta > 1.
(a) Show that log Xi has an exponential distribution with a mean of 1/theta.
(b) Find the form for a UMP test of H_0: theta <= theta_0 vs. H_a : theta > theta_0.
(c) Give formula for nding the rejection region for a given value of alpha.
Hint: use the result from (a) to find the distribution of the test statistic.
(d) Conduct the test in (b) with alpha = 0.05 and theta_0 = 1:5 using the dataset: {1.2, 2.4, 1.3, 1.7, 1.9}.
(e) Find the form of a UMPU test for testing H_0 : theta = theta_0 vs. H_a : theta_0 != theta_0.
(f) Use the data in (d) to conduct the test in (e) with alpha = 0.05 and theta_0 = 1.5.

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- 1); I{x>1}, theta > 1.
(a) Show that log Xi has an exponential distribution with a mean of 1/theta.
(b) Find the form for a UMP test of H_0: theta <= theta_0 vs. H_a : theta > theta_0.
(c) Give formula for nding the rejection region for a given value of alpha.
Hint: use the result from (a) to find the distribution of the test statistic.
(d) Conduct the test in (b) with alpha = 0.05 and theta_0 = 1:5 using the dataset: {1.2, 2.4, 1.3, 1.7, 1.9}.
(e) Find the form of a UMPU test for testing H_0 : theta = theta_0 vs. H_a : theta_0 != theta_0.
(f) Use the data in (d) to conduct the test in (e) with alpha = 0.05 and theta_0 = 1.5.

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- 1); I{x>1}, theta > 1.
(a) Show that log Xi has an exponential distribution with a mean of 1/theta.
(b) Find the form for a UMP test of H_0: theta <= theta_0 vs. H_a : theta > theta_0.
(c) Give formula for nding the rejection region for a given value of alpha.
Hint: use the result from (a) to find the distribution of the test statistic.
(d) Conduct the test in (b) with alpha = 0.05 and theta_0 = 1:5 using the dataset: {1.2, 2.4, 1.3, 1.7, 1.9}.
(e) Find the form of a UMPU test for testing H_0 : theta = theta_0 vs. H_a : theta_0 != theta_0.
(f) Use the data in (d) to conduct the test in (e) with alpha = 0.05 and theta_0 = 1.5.

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lives. Suppose you are in class of 52 students.
A) what is the probability that over 13 students experience stress
B) if 23 students in the class said they felt stress in their daily lives would you be surprised
Suppose the average ACT score for students taking the test in Illinois is in 2003 was 21.8 with a standard deviation of 3.85. What is the probability that 32 randomly selected students from that state average under 23 on the ACT that year?

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lives. Suppose you are in class of 52 students.
A) what is the probability that over 13 students experience stress
B) if 23 students in the class said they felt stress in their daily lives would you be surprised

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2,3,4,7). If it lands tails, a fair six-sided die is
thrown (with values 3,4,5,6,7,9). Regardless of which die is used, Alice
eats n grains of rice, where n is the largest prime factor of the die result
(for example, the largest prime factor of 9 is 3).
(a) What is the conditional probability that the coin lands heads, given
that Alice eats three grains of rice?
(b) Suppose that the entire experiment is conducted twice on the following day (starting with a new coin toss on the second run-through).
What is the conditional probability that the coin lands heads on both
run-throughs, given that Alice eats a total of five grains of rice during the two run-throughs?
(Do not count the two grains from part (a) in part (b); we assume
two brand new experiments, each with a new coin toss. Start your
solution by defining a suitable partition of the sample space. Please
use an appropriate notation and/or justification in words, for each
value that you give as part of your solution.)
Exercise 5) Alice and Bob throw an unfair coin repeatedly, with probability 2/5 of landing heads. Alice starts with £2 and Bob starts with £3 .
Each time the unfair coin lands heads, Alice gives Bob £1 . Each time
the unfair coin lands tails, Bob gives Alice £1 . The game ends when one
player has £5 .
(a) Draw a labelled Markov chain describing the problem, and write
down a transition matrix P. Write down the communication classes,
and classify them as either recurrent or transient.
(b) Using the transition matrix, calculate the probability that Alice loses
all of her money in exactly four tosses of the unfair coin.
(c) Calculate the (total) probability that Alice loses all of her money
(before Bob loses all of his).
(d) Calculate the expected (mean) number of tosses of the unfair coin,
for the game to end.

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a prize with probability p independently of the other curtains. The player chooses one curtain
(a) The game host tells the contestant there's a prize hiding behind at least one of the other curtains, and offers the player a chance to change his choice. Should he?
(b) The game host tells the player there's a prize hiding behind exactly one of the other curtains. Should the player change his choice? How does your answer depend on p
(c) How would your answer to section (b) change if there are n curtains in total?
(d) Bonus (4 points): How would your answer to section (a) change if there are 4 curtains in total?

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tosses (i.e. the second, fourth, sixth etc.) are heads?
(b) In an infinite series of independent coin tosses, what is the probability that the first heads is followed by a tails?

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28.Amy has 8 cloth masks in 4 different colors- 2 of each. She picks a mask daily at random. In ...

2 masks are not in the pile (used/laundry)
(a) What is the probability Amy picks a mask in her favorite color?
(b) Given that yesterday she drew her favorite color, how would your answer to (a) change?

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of cans that are inspected before one with a bump or dent is found.
What is the probability that more than 3 cans are inspected before one with a bump or dent is found?

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two objects and recorded whether or not the dog being tested correctly chose the object indicated. A four-year-old male beagle named Augie participated in this study. He chose the correct object 42 out of 70 times when the experimenter leaned towards the correct object.
(a) (2 points) Let the parameter of interest, π, represent the probability that the long-run probability that Augie chooses correctly. Researches are interested to see if Augie understands human body cues (better than gussing).
Fill in the blanks for the null and alternative hypotheses.
H0 : Ha :
(b) (6 points) Based on the above context, conduct a test of significance to determine the p-value to investigate if domestic dogs understand human body cues. What conclusion will you draw with significance level of 10%? (If you use an applet, please specify which applet you use, and the inputs.)
(c) (5 points) Based on the above context, conduct a test of significance to determine the p-value to investigate if domestic dogs understand human body cues. What conclusion will you draw with significance level of 5%? (If you use an applet, please specify which applet you use, and the inputs.)
(d) (2 points) Are your conclusions from part (b) and (c) the same? If they are different, please provide an explanation.
(e) (5 points) Shown below is a dotplot from a simulation of 100 sample proportions under the assump- tion that the long-run probability that Augie chooses correct is 0.50. Based on this dotplot, would a 90% confidence interval for π contain the value 0.5? Explain your answer.
(f) (4 points) Compute the standard error of the sample proportion of times that Augie chose the object correctly.
1
(g) (5 points)
(h) (3 points) question?
(i) (4 points)
(j) (4 points) A.
B. C.
Construct an approximate 95% confidence interval for π using the 2SD method. What is the margin of error of the confidence interval that you found in the previous
How would you interpret the confidence interval that you found in part (g)?
Which of the following is a correct interpretation of the 95% confidence level?
If Augie repeats this process many times, then about 95% of the intervals produced will capture the true proportion of times of choosing the correct objective.
About 95% times Augie chooses the correct objective.
If Augie repeats this process and constructs 20 intervals from separate independent sam- ples, we can expect about 19 of those intervals to contain the true proportion Augie chooses the correct objective.
(k) (4 points)
object 21 out of 35 times.
Conjecture how, if at all, the center and the width of a 99% confidence interval would change with these data, compared to the original 2SD 95% confidence interval.
The center of the confidence interval would . The width of the confidence interval would .
(l) (4 points) Suppose that we repeated the same study with Augie, and this time he chose the correct object 17 out of 35 times, and we also change the confidence level from 95% to 99%. Conjecture how, if at all, the center and the width of a 99% confidence interval would change with these data, compared to the original 2SD 95% confidence interval.
Suppose that we repeated the same study with Augie, and this time he chose the correct
The center of the confidence interval would The width of the confidence interval would
.
.

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31.hI I need help with a FINITE MATH question. In a survey of high school graduates, it was found that ...

if a graduate went on to college full time, there was a 23% chance that they would work at the same time. If a graduate attended college part time, there was a 79% chance that they would work at the same time. If a high school graduate didn't go on to college, there was a 93% chance that they would work. At the time of the survey, 66% of high school graduates attended college full time and 20% of high school graduates attended college part time. If a randomly selected high school graduate worked, what is the probability that they attended college full time? Enter your answer in decimal form and round to four decimal places if necessary. You may want to look at pages 42 through 45 in the notes (videos 30 through 32).
If a randomly selected high school graduate worked, probability that they attended college full time =

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’s left side.
a) (8 points) Among 20 randomly selected students at a university, what is the probability that less than 3 of them have left-brain language control?

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population means are equal

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tion of X is given as follows:
X 3 4 6
p(X) 0.3 0.4 0.3
Find mean and variance of distributions X, Y.

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ikes 8 of the cabins and 2 of the cafeterias. What is
the probability that he is assigned to both a cabin
and a cafeteria that he likes?

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sample of 175 KU students, what is the probability that this sample will yield a sample proportion less than 0.30

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37.In a bag there is 12 red,8 yellow and 5 green balls. Bill takes upp balls without looking what is ...

e probability that she gets
2 red balls
3 balls of the same color

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members of the class to be in a video you’re making, but at least 2 must be
seniors. How many different ways are there to choose the students?

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e customer is 4 per year with the standard deviation 2. Find the probability that the total number of yearly claims in the service department does not exceed 50000.

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nd standard deviation 6 inches.
A button hyperlink to the SALT program that reads: Use SALT.
(a) What is the probability that an 18-year-old man selected at random is between 64 and 66 inches tall? (Round your answer to four decimal places.)
Correct: Your answer is correct.
(b) If a random sample of seven 18-year-old men is selected, what is the probability that the mean height x is between 64 and 66 inches? (Round your answer to four decimal places.)
Incorrect: Your answer is incorrect.
(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?
The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.
The probability in part (b) is much higher because the standard deviation is larger for the x distribution.
The probability in part (b) is much higher because the mean is smaller for the x distribution.
The probability in part (b) is much higher because the mean is larger for the x distribution.
The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.

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average rate of 1.1% since then, how many years later would the population arrive at 102.000 people? Give your answer as a number only, to the nearest tenths
Question 3
In a bag containing 5 red, 2 blue, 8 clear, and 1 purple marble, what is the probability of picking a blue marble, then without replacement, choosing a red marble on the second pick? (Leave in simplified fraction form.]
Question 11
Poaching is causing a population of elephants to decline by 6% per year. Determine how many elephants remain in 76 years if there are 13,365 elephants today. Round to the nearest whole number.

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it sound like a math problem.
The problem :
What are the chances of a 4 sided die landing on 1 twice and on 2 twice out of 4 rolls. The solution I came up with originally was (2/4) x (2/4) x (2/4) x (2/4) . Which I realized was wrong as this allowed the die to land on 1 four times in a row. So then I came up with this soultion (which i still think is wrong) (2/4) x (2/4) x (1/4) x (1/4) . So the reasoning behind this is : The first roll obviously has a 50% chance to roll on either 1 or 2. Second roll is the same. BUT, lets say both of them land on 1, and now it HAS to land on 2 the remaining two times. So my problem is with the current solution that I have is what if the die lands on 1 on the first roll, then on two for the second one. then the third roll would still have a 2/4 AKA a 50% chance of landing on either one. I'm sure the last roll is 1/4 but I just dont know if the order matters on the rolls. This has been driving me crazy the last hour. Please help if you can thanks

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6 minutes and the variance of the waiting time is 9. Find the probability that a person will wait for between 1 and 9 minutes. Round your answer to four decimal places.

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es them to perform a task. Assuming that all of the times are different, how many different rankings are possible?"

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tests, probability distributions, regression/correlation and categorical data analysis will all be covered)
We are going to be given a variety of questions. I am having doubt in computing the correct numbers on excel. Eg: I'll be 1-2 decimal places behind. I am attaching a formula sheet.

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with a mean life of 3000 hours and a standard deviation of 400 hours.
What is the probability that a randomly selected light bulb last for more than 4000 hours?

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47.In a jury pool selection, 65% of the people are female. Of those 65%, 25% of them work in the ...

care field.
Find the probability that a randomly selected person from the jury pool is female and works in the health field.
Find the probability that a randomly selected person from the jury pool is female and does NOT work in the healthcare field

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l-time freshmen from 270 four-year colleges and universities in the U.S. 71.2% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same-sex couples should have the right to legal marital status.
Construct the probability distribution function (PDF). (Round your probabilities to five decimal places.)

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lub.
The probability of being dealt a club is

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ibuted. The average distance is 13.8 miles. The standard deviation is 3.2 miles. Use the empirical rule to estimate the probability of a victim traveling longer than 7.4 miles.
Answer choices:
0.15%
97.5%
2.5%
99.85%

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apan averaging 89 years with a standard deviation of 4 years, normally
distributed. In Tokyo prefecture, female life expectancy averages 85 years with a
standard deviation of 5 years, normally distributed.
(a) What is the probability that a woman moving from Okinawa to Tokyo prefecture will
result in higher life expectancy in both Okinawa and Tokyo?
(b) What is the probability that a group of four women moving from Tokyo to Okinawa
will result in lowering life expectancy in both prefectures?

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robability that a parachute will fail to open is 0.0025 and 0.002, depending on whether it is from company C1 or C2, respectively.
a) What is the probability that a randomly chosen parachute will fail to open (rounded off to four decimals)?
b) If a randomly chosen parachute fails, what is the probability it came from company C1 (rounded off to four decimals)?
c) If a randomly chosen parachute fails, what is the probability it came from company C2 (rounded off to four decimals)?

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cally b) and c) please?
The Air force receives 30% of its parachutes from company C1 and the rest from company C2. The probability that a parachute will fail to open is 0.0025 and 0.002, depending on whether it is from company C1 or C2, respectively.
a) What is the probability that a randomly chosen parachute will fail to open (rounded off to four decimals)?
Ans=0.0022
b) If a randomly chosen parachute fails, what is the probability it came from company C1 (rounded off to four decimals)?
c) If a randomly chosen parachute fails, what is the probability it came from company C2 (rounded off to four decimals)?

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on a subsample of weekly data from Randall’s Supermarket, one of the biggest in the UK. Randall’s marketing management team wishes to identify trends and patterns in a sample of weekly data collected for a number of their loyalty cardholders during a 26-week period. The data includes information on the customers’ gender, age, shopping frequency per week and shopping basket price. Randall’s operates two different types of stores (convenient stores and superstores) but they also sell to customers via an online shopping platform. The collected data are from all three different types of stores. Finally, the data provides information on the consistency of the customer’s shopping basket regarding the type of products purchased. These can vary from value products, to brand as well as the supermarket’s own high-quality product series Randall’s Top. As a business analyst you are required to analyse those data, make any necessary modifications in order to determine whether for any single customer it is possible to predict the value of their shopping basket.
Randall’s marketing management team is only interested in identifying whether the spending of the potential customer will fall in one of three possible groups including:
• Low spender (shopping basket value of £25 or less)
• Medium Spender (shopping basket value between £25.01 and £70) and
• High spenders (shopping basket greater than £70)
For the purpose of your analysis you are provided with the data set Randall’s.xls. You have to decide, which method is appropriate to apply for the problem under consideration and undertake the necessary analysis. Once you have completed this analysis, write a report for the Randall’s marketing management team summarising your findings but also describing all necessary steps undertaken in the analysis. The manager is a competent business analyst himself/herself so the report can include technical terms, although you should not exceed five pages. Screenshots and supporting materials can be included in the appendix.
Requirements
After completing your analysis, you should submit a report that consists of two parts. Part A being a non-technical summary of your findings and Part B a detailed report of the analysis undertaken with more details.
Part A: A short report for the Head of Randall’s Marketing Management (20 per cent). This should briefly explain the aim of the project, a clear summary and justification of the methods considered as well as an overview of the results.
Although, the Head of Randall’s Marketing Management team who will receive this summary is a competent business analytics practitioner, the majority of the other team members have little knowledge of statistical modelling and want to know nothing about the technical and statistical underpinning of the techniques used in this analysis. This report should be no more than two sides of A4 including graphs, tables, etc. In this report you should include all the objectives of this analysis, summary of data and results as well as your recommendations (if any).
Part B: A technical report on the various stages of the analysis (80 per cent).
The analysis should be carried out using the range of analytics tools discussed:
• SPSS Statistics
Ensure that the exercise references:
• Binary and multinomial logistic regression
• Linear vs Logistic regression
• Logit Model with odds Ratio
• Co-efficients and Chi Squared
• MLR co-efficients
• Assessing usefulness of MLR model
• Interpreting a model
• Assessing over-all model fit with Psuedo R-Squared measures
• Classification accuracy (Hit Ratio)
• Wald Statistic
• Odd ratio exp(B)
• Ratio of the probability of an event happening vs not happening
• Ratio of the odds after a unit change in the predictor to the original odds
• Assumptions
• Residuals analysis
• Cook’s distance
• DfBeta
• Adequacy (with variance inflation factor VIF and tolerance statistic)
• Outliers and influential points cannot just be removed. We need to check them (typo? – unusual data?)
• Check for multicollinearity
• Parsimony
Write a short and concise report to explain the technical detail of what you have done for each step of the analysis.
The report should also cover the following information:
• Any type of analysis that might be useful and check whether the main assumptions behind the analyses do not hold or cannot be
• Give evidence of the understanding of the statistical tools that you are using. For example, comment on the model selection procedure and the coefficient interpretation, e.g. comment on the interpretation of the logistic regression coefficients if such a method is used and provide an example of
• Conclusions and explanation, in non-technical terms, of the main points

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56.* Fill in the blanks. Example: If I say that at a 95% confidence level and a 5% margin of ...

there are 33% pink marbles in the bag, I’m saying that there is a 95% probability (or, 95 times out of 100) that there are between 28% and 38% pink marbles.
99% confidence level, 10% margin of error
=> There is a __% probability (or, __ times out of 100) that there are between __% and __% pink marbles.
90% confidence level, 7% margin of error
=> There is a __% probability (or, __ times out of 100) that there are between __% and __% pink marbles.

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Leaf and 6 of the employees are vegetarian and own a Nissan Leaf. If you randomly select an employee from company XYZ, what is the probability that the employee is vegetarian or that she/he owns a Nissan Leaf?

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is the probability of an individual phone being defective? If a quality control engineer wants to carefully analyze a defective mobile phone, what is the probability of her getting at least one defective phone in a lot of 25? Is the probability high enough that the engineer can be reasonably sure of getting a defective mobile phone that can be used for her analysis?

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lity...
"As part of a probability experiment, Elliott is to answer 4 multiple-choice questions. For each question, there are 3 possible answers, only 1 of which is correct. If Elliott randomly and independently answers each question, what is the probability that he will answer the 4 questions correctly?"
Solving the question I keep getting 1/256. My reasoning is that each question has a 1/4 chance of being right if you guessed, so 1/4 x 1/4 x 1/4 x 1/4 = 1/256.
But my answer choices are as follows, with the correct answer being E:
A. 27/81
B. 12/81
C. 4/81
D. 3/81
E. 1/81
Am I missing something? Or am I just completely solving this wrong?

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de by credit card; you think this is true at your store as well. On a typical day you make 20 sales.
Show that this situation can be modeled by a binomial distribution. For credit, you must discuss each of the criteria required for a binomial experiment.
Define the random variable x in this scenario, using the context of the problem.
List all possible values of x for this situation.
On one trial for this scenario, what does “success” mean? Explain using the words of the problem.
What is the probability of success in this scenario?
What is the probability of failure in this scenario?
Probability Distribution Instructions¬¬¬¬
In Excel, create a probability distribution for this scenario.
Label Column A as “x” and Column B as “P(x).”
In Column A, list the numbers 0 to 15.
In Column B, use BINOM.DIST.RANGE to calculate the probability for each x value.
Highlight the probability cells, then right click and select Format Cells. Format the probability cells as “Number” and have Excel show 4 decimal places.
Create a probability histogram using the probabilities you calculated. Format and label it properly. Be sure to use the “Select Data” button to change the x-axis so it correctly lists the x-values.

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the probability that within the 60 selected children 18 to 20 of them call their families daily.

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ose
the breaking point is uniformly distributed on the chalk. Let X denote the length of the shorter piece and
R the ratio of the lengths of the shorter to the longer piece. Then X has uniform distribution on [0; 1
2 ].
(a) (6pts) Find the probability density function of R.
(b) (8pts) Find the mean and variance of R.

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certain casino, the average amount lost by a random gambler is $20
per night, with a standard deviation of $135. If we randomly select 60 gamblers, find the
probability that as a group they win more than they lose that night.dard deviation of $135. If we randomly select 60 gamblers, find the
probability that as a group they win more than they lose that night.

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deviation of $135. If we randomly select 60 gamblers, find the
probability that as a group they win more than they lose that night.

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66.A data set has a mean of 85 and a standard deviation of 4 and is distributed normally. What is ...

robability that a randomly selected data value is less than 72?

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67.Vanessa found that the average for all students who feel that good manners are necessary to ...

75 with a sample standard deviation of 15; (this data is normally
distributed). If she took a random sample of 64 students that shared this
view
Find:
a) The probability that the average number of students that share this view is more than 78 students.
b) The probability that the mean number of students that share this view is
between 65 and 72

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68.Twenty students are asked to select an integer between 1 and 10. Eight choose either 4, 5 or 6. a If ...

f the students make their choices independently and each is as likely to pick one integer
as any other, what is the probability that 8 or more will select 4,5 or 6?
b Having observed eight students who selected 4, 5, or 6, what conclusion do you draw based
on your answer to part (a)?A missile protection system consists of n radar sets operating independently, each with a
probability of .9 of detecting a missile entering a zone that is covered by all of the units.
a If n = 5 and a missile enters the zone, what is the probability that exactly four sets detect
the missile? At least one set?
b How large must n be if we require that the probability of detecting a missile that enters the
zone be .999?

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of 45 inches and a standard deviation of 8 inches. What is the probability
a randomly-selected student is taller than 50 inches?

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100 and a standard deviation of 15.
a. What is the probability that a given applicant will score over 100?
b. Determine the probability that a random applicant scores over 135.
c. Determine the probability that an applicant scores between 85 and 135.

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71. The probability of a success is 0.2 for each of 10 trials. What is the probability that one gets ...

actly 3 successes and 7 failures if each trial is independent of the others? What is the probability that only the first, third and fifth trials are successes and the rest are failures?
3 One is dealt 6 cards from a standard poker deck of 52 cards. What is the probability of getting 3 aces and 3 kings? What is the probability of getting the same except both pairs kings and aces are of the same suit (e.g., the same suit is missing from both).

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cookies and 5 oatmeal raisin cookies. What is the probability that Noah randomly selects a peanut butter cookie from the bag, eats it, then randomly selects another peanut butter cookie?

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IRS employee randomly selects and checks 10 forms for mistakes. What is the probability that exactly one has an error?

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consumes each year is 196 with a standard deviation of
22 pounds (Source: American Dietetic Association). If a sample of 50 individuals is randomly selected, find
the probability that the mean of the sample will be less than 200 pounds.

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ng n distinct integers in just his thoughts.
He plays turns on the list. In each turn he does the following–
He takes a number (in its index’s order ) and swap it with any number in
the list including itself i.e. if it swap it with itself it doesn’t move at all (The
selection of the number is completely random).
He does the same for all the elements in their index’s order in that turn.
If initially the list was unsorted, such that, no element was in sorted position,
then find the probability that the list is sorted after m such turns.
Note : Take the assumption that if an element is not in its sorted
position then it can be in any other n − 1 positions equally likely.

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bers 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?

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a weekday (Monday through
Friday)

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79.Hi! I need help with part (c) of this question. I have already done part (a) & (b): A recent poll ...

conducted by Gallup Inc. asked the following question:
In general, how much trust and confidence to you have in the mass media - such as newspapers, TV, and radio - when it comes to reporting the news fully, accurately, and fairly? A great deal, a fair amount, not very much, or non at all?
53% of respondents has a negative view of the media, meaning they responded with either not very much or non at all.
Two people are randomly chosen, each is asked the poll question above. What is the probability
Part (a) that both have a negative view of mass media? 0.53^2
Part (b) neither have a negative view of the mass media? 0.47^2
Part (c) at least one of the two has a negative view towards the media? .53 (<- wrong)

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ut, triple chocolate, cinnamon sugar, chocolate chip, and peanut butter. If you must choose at least one cookie: (Answer to one decimal place)
What is the probability that you do not take cookies that contain chocolate? Evaluate as a percent to the nearest 1 decimal place if rounding is required.

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81.Alice has two coins. The probability of Heads for the first coin is 1/4 , and the probability of ...

ads for the second is 3/4 . Other than this difference, the coins are indistinguishable. Alice chooses one of the coins at random and sends it to Bob. The random selection used by Alice to pick the coin to send to Bob is such that the first coin has a probability

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hey are receiving. what is the probability that a patient selected at random from this offices patients is a male who is satisfied with the care he is receiving.

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ere 45 Republicans and 55 Democrats*. Use this information to answer the following.
*Technically, 2 of these Senators were Independents or Independent Democrats, but caucused with the Democratic Party.
Remember the rounding convention for a decimal value with more than 3 zeroes is to round to the second non-zero digit.
If we choose a committee of 10 at random, what is the probability that they will all be Republicans?

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e forecast calls for a 70% chance of snow on Saturday and a 50% chance of snow on Sunday. What is the probability that it will snow both days?
I think the answer is 35% but my teacher disagrees. (0.7 x 0.5 = 0.35)
Can you help me?

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e forecast calls for a 70% chance of snow on Saturday and a 50% chance of snow on Sunday. What is the probability that it will snow both days?
I think the answer was 35% but my teacher disagrees. Can you help me?

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87.The probability that a group of people contains at least one person with an IQ in the top 1% goes ...

the larger the group gets. At what number of people does it become more likely than not that there will be at least one 1%er?

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debit card and 20.3% have both. what is the probability that a participant has KU wireless or a debit card?

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h of them. The probability that he makes a mistake in his
thinking is 0.005. The probability that an assistant gives wrong information is 0.3.
Assuming that the mistakes made by the manager are independent of the
information given by the assistants, find the probability that he reaches a wrong
decision.

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y of getting "x" candy inside the box that can only hold 20 candies, then getting that specific candy("x" candy)?
Not sure if that make sense

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A (refresher), linear regression, logistic regression, multivariate regression, chi square test (refresher), and more. CLT is a given, so are probability distributions;
I eat MC simulations for breakfast. I only work in Excel; not so much interest in Crystal Ball and other add-ins; I have created my own VBA Excel functions such as Spearman(), triangularprobabilitydist() and PoissonInv().
So hit me with what you got, I am up to the challenge. looking for some fun and challenging time.
Availability evenings and weekends; looking at 20+ hours of tutoring over next 6 weeks.

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deo a plus) connection and use of the whiteboard. I am on Central European Time Zone (GMT+1).
Topics: Continuous Probability Distributions, Assessing normality, Sampling Distribution and the Central Limit Theorem

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ppose a package of M&M’s typically contains 52
M&M’s.
a.) State the random variable.
b.) Argue that this is a binomial experiment
Find the probability that
c.) Six M&M’s are brown.
d.) Twenty-five M&M’s are brown.
e.) All of the M&M’s are brown.
f.) Would it be unusual for a package to have only brown M&M’s? If this were
to happen, what would you think is the reason?

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ence for hamburger or chicken. Of 200 respondents selected, 125 were male, 75 were female. 120 preferred hamburger and 80 preferred chicken. Of the males, 85 preferred hamburger. Suppose that two individuals are randomly selected. The probability that both prefer hamburger is:
Question 8 options:
a)
9/25
b)
3/5
c)
357/995
d)
17/40

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weighing between 140 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 145 lb and a standard deviation of 29.9 lb.
a. If a pilot is randomly selected, find the probability that his weight is between 140 lb and 191 lb.

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et loans (he doesn't "earn" income). With the loans (L) he needs to decide between first period consumption (C1) and investment (I). The amount invested will allow him to get a second period income Y with probability P which is increasing in I (therefore, P(I)), In case of success and the person obtain Y, the individual should use Y to repay the loan (L) that he requested in the first period and consume in the second period (C2). However, with probability 1 - P(I), the person don't get Y and therefore only consume C1. Note that if the individual only invest the loan (L=I) and don't obtain Y, he can't consume anything. That motivates him not to invest the whole loan and keep part of the loan in order to warrant at least first period consumption. Therefore, considering B the parameter for the time preference, the problem would be:
max U=ln(C1) + Bln(C2)
s.t: L = I + C1
Y(I) = L(1+r) + C2
with Probability P(I)
or
s.t: L = I + C1
with probability 1 - P(I)
My question is, Have you ever seen something like this? If yes, how to proceed? What is more important, I really need a bibliography (a book or article talking about this)

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selected, what is the probability that the subject is not lying? Is the result close to the probability of 0.395 for a negative test result?
Did the Subject Actually Lie?

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36 Using your knowledge of probability, why should Nizar have known that his answer was NOT correct and gone back to review his calculations?
2. A four-colored spinner was spun 80 times. The spinner landed on green 15 times, yellow 30 times, blue 10 times, and red 25. Based on this data, what is the experimental probability of the spinner landing on green? (Write your answer as a fraction in lowest terms.)
3. If the probability that a certain mechanical part in your new car will fail this year is 0.05, what is the probability that the mechanical part will not fail this first year?
4. A coin is loaded so that the probability of getting tails is 1/4. If the coin is flipped twice, what is the probability of getting tails twice?
5.If a 6-sided die is tossed and then a coin is flipped, what is the probability that an odd number is rolled and the coin lands on heads?
6.A field goal kicker makes 3 of every 7 attempts at a field goal. If he kicks 4 field goals in a certain game, what is the probability that he'll make all four? (Write your answer as a fraction in lowest terms.)

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99. Value: 1 equation image indicator a. (x - 2)2(x - 3)2 b. (x2+ 4)(x2+ 9) c. (x - 2)(x + ...

2)(x - 3)(x + 3)
d. (x2 - 4)(x2+ 9)
Value: 1
The table below shows the cost of purchasing a standard stapler at five office supply stores, A through E. If the median cost of purchasing a standard stapler for these stores was $17.99, which of the following could NOT have been the cost of the stapler for Store A?
staplergraph.jpg
a. $19.95
b. $18.95
c. $16.95
d. $19.25
Value: 1
If equation image indicator then x =
a. 7
b. 1/5
c. 5
d. 1/7
Value: 1
A six−sided die, with sides numbered 1,2, 3,4,5, and 6, is tossed. What is the probability of tossing a number less than three?
a. 1/3
b. 0
c. 1/2
d. 1/4
Value: 1
If 6m + 4 = 8m, then 4m =
a. 6
b. 2
c. 8
d. 4
Value: 1
In the xy-plane, what is the y-intercept of the graph of the equation equation image indicator?
a. 2
b. 4
c. 16
d. There is no y-intercept.
Value: 1
Which of the following equations has both 2 and −4 as solutions?
a. x2 + 6x + 8 = 0
b. x2 - 2x - 8 = 0
c. x2 + 2x - 8 = 0
d. x2 - 2x + 8 = 0
Value: 1
The perimeter of a square is 20 ft. If you increase the length of the square by 2 feet and decrease the width by 1 foot, what is the area, in square feet, of the new figure?
a. 22
b. 28
c. 35
d. 40
Value: 1
(3x-2y4)-3 =
a. equation image indicator
b. equation image indicator
c. equation image indicator
d. equation image indicator
Value: 1
A softball is tossed into the air upward from a first floor balcony. The distance of the ball above the ground at any time is given by the function, distance function.png, where h(t) is the height of the softball above the ground (in feet) and t is the time (in seconds). What was the maximum height, in feet, of the softball above the ground after it was thrown?
a. 28
b. 30
c. 32
d. 34
Value: 1
A group of 100 people, some students and some faculty, attended a museum opening. Each student paid $10 per person for entrance to the museum and each of the faculty paid $25 per person for entrance. If the total paid, for all 100 people, was $1300, how many students attended the museum opening?
a. 20
b. 50
c. 70
d. 80
Value: 1
The ratio of Sam's age to Hank's age is 5 to 3. If the sum of their ages is 24, how old is Hank?
a. 21
b. 15
c. 19
d. 9
Value: 1
In the xy−coordinate plane shown below, point P has coordinates (8, −6). Which of the following is an equation of the line that contains points O and P?
O and P graph.jpg
a. equation image indicator
b. equation image indicator
c. equation image indicator
d. equation image indicator
Value: 1
The variables x and y are inversely proportional, and y = 2 when x = 3. What is the value of y when x = 9?
a. 54
b. 6
c. 2/3
d. 3/2
Value: 1
A farmer has 1235 trees to be planted on a rectangular parcel of land. If there are 24 trees planted in each row and each row must be complete before it is planted, how many trees will be left over after planting?
a. 21
b. 11
c. 0
d. 55

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oice question is 60%, use the binomial tables to determine the probability of earning at least a 70% grade on a 20-question exam
What is the probability of earning at least a 70%

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:
Given, P(A)=0.85, P(A ∩ B')=0.50
a.) What is P(A∩B)?
b.) What is P(B'|A')?

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f a particular brand of jelly available, 84 preferred strawberry, 37 preferred grape, and 100 people would buy one or the other or both. Fill in the remainder of the two-way frequency table.
Strawberry Not Strawberry TOTAL
Grape 37
Not Grape 50
TOTAL 84 150
What is the probability (rounded to the nearest whole percent) that a randomly selected shopper would buy both grape jelly and strawberry jelly?

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is a multiple of three or greater than eight.
A certain game consist of rolling a single fair die and pays off as follows nine dollars for a six, six dollars for a five, one dollar for four and no payoffs otherwise.Find the expected winnings for this game.
A fair die is rolled four times. A 6 is considered success While all other outcomes are failures find the probability of three successes.
A pet store has nine puppies including 4 poodles 3 terriers and 2 retrievers. If Rebecca an errand in that order each select one puppy at random without replacement find the probability that Aaron select a retriever given that from last Rebecca selects a poodle.
Experience shows that a ski lodge will be for (166 guests) if there is a heavy snowfall in December, well only partially full (52 guests) With a light snowfall. What is the expected number of guests if the probability for a heavy snowfall is 0.40? I assume that heavy snowfall and light snowfall are the only two possibilities.
A pet store has six puppies Including two poodles two Terriers and to retrievers. If Rebecca and Aaron in that order each select one puppy random with replacement (They both may select the same one) Find the probability That Rebecca selects a terrier and Aaron selects a retriever.
Three married couples arrange themselves randomly in six consecutive seats in a row. Determine (A) the number of ways the following event can occur, And (B) the probability of the event. (The denominator of the probability fraction will be 6!=720, The total number of ways to arrange six items ). Each man was that immediately to the right of his wife.
A coin is tossed five times. Find the probability that all our heads. Find the probability that at least three are heads.
A certain prescription drug is known to produce undesirable facts and 35% of all patients due to drug. Among a random sample of a patient using a drug find the probability of the stated event. Exactly 5 have undesired effects.
10,000 raffle tickets are sold. One first prize of 1600, for second prizes of 800 each, And 9/3 prizes of 300 each or to be awarded with all winners selected randomly. If you purchase one ticket what are your expected winnings.
Suppose a charitable organization decides to Raise money by raffling A trip worth 500. If 3000 tickets are sold at one dollar each find the expected net winnings for a person who buys one ticket. Round to the nearest cent
Three men and seven women are waiting to be interviewed for jobs. If they are selected in random order find the probability that all men will be interviewed first
A fair diet is rolled. What is the probability of rolling on our number or a number less than three.
The pet store has 15 puppies, including five poodles, five Terriers, and five retrievers. If Rebecca and Aaron, in that order, select one puppy at random without replacement, find the probability that both select a poodle
Beth is taking a nine question multiple-choice test for which each question Has three answer choices, only one of which is correct. Beth decides on answering By rolling a fair die And making the first answer choice if the die shows one or two, The second If the die shows three or four, and the third if the die shows five or six. Find the probability of the stated event. Exactly 6 correct answers
For the experiment of drawing a single card from a standard 52 card deck find (a) the probability and (b) the odds are in favor that they do not drive six

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104.Question 1: suppose the dealer has an an Ace showing, you have a 3 and an 8 in your hand, ...

r player at the table is showing a 5 and 6. Compute the expected value of a one-dollar insurance bet under these circumstances
Question 2: Suppose that( perhaps after being hit one or more times) you have cards addings up to 18. Using the table provided in these notes, compute the probability that you will lose, and the probability that you will tie.
Question 3: The "Royal Hand" consists of King and Queens of the same suit. Compute the probability of being dealt a Royal Hand in the first two cards.
Question 4: Compute the probability that you will initially be dealt two cards adding up to exactly 20. ( First think about how many ways two cards can up to 20 in blackjack.)
Question 5: You have two 9s in your hand. The dealer is showing a 7, and the only other player at the table is sowing a King and a 9. If you ask to be hit, what is the probability that you will bust on the next card?

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of the complement of Event B?

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with a mean of 25 gm and a standard deviation of 5 gm.
(a) If the machine is used 500 times, approximately how many times will it be expected to dispense 30 gm or more of chilli sauce?
(b) How can you decrease this number to half? Give a numerical answer.
2. StarTech manufactures re sensors. They use a protective screen for their sensors to protect it from dust. The sensor becomes useless if the thickness of the screen exceeds 0.5 mm. They outsource the production of the screen to a di erent company that claims to manufacture screens with a mean thickness of 0.3 mm and a standard deviation of 0.1 mm.
(a) If 10000 screens are manufactured how many will be discarded because they are too thick?
(b) If screens less than 0.2 mm are too thin to be used, what is the probability that screens manufactured by the above company will be discarded because they are too thick or too thin? Show the result on a graph.
3. The amount of time that Sam spends playing the guitar is normally distributed with a mean of 15 hours and a standard deviation of 3 hours.
(a) Find the probability that he spends between 15 and 18 hours playing the guitar during a given week.
(b) What is the probability that he spends less than 3 hours playing the guitar during a given week?
4. Soon after he took oce in 1963, President Johnson was approved by 160 out of a sample of 200 Americans. With growing disillusionment over his Vietnam policy, by 1968 he was approved by only 70 out of a sample of 200 Americans.
(a) What is the 90% con dence interval for the percentage of all Americans who approved of Johnson in 1963? In 1968?
(b) What is the 90% con dence interval for the change?

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107.Let's say that I have a data set of every time a certain person has coughed since the beginning of ...

his year. I'm looking for a formula that will tell me the probability that a person will cough x number of times in a given week.
I started with the Poisson Distribution, but Poisson doesn't seem to take into account standard deviation. To calculate the probability with Poisson, only the mean, expected value, and test value are needed, meaning the variance/standard deviation of the data could vary widely, and you'd still get the same probability distribution. For example, if someone coughed exactly 5 times everyday, you'd get the same probability distribution if this person alternated coughing 0 times one day, 10 times the next, 0 times the next day, 10 times the next, and so on.
Does my question make sense? Thanks for your help.

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1.AU MAT 120 Systems of Linear Equations and Inequalities Discussion

mathematicsalgebra Physics