# Let D be the set of all students at your school, and let M(s) be a ”s is a math major”, let C(s)”s is a computer science student”, and let E(s) be ”s is an engineering student.” Express each of the following statements using quantifiers, variables, and predicates M(s), C(s) and E(s) Question
Probability and combinatorics Let D be the set of all students at your school, and let M(s) be a ”s is a math major”, let C(s)”s is a computer science student”, and let E(s) be ”s is an engineering student.” Express each of the following statements using quantifiers, variables, and predicates M(s), C(s) and E(s) 2021-01-05
Solution:
$$\displaystyle{D}=$$ set of all students at your school (domain)
$$\displaystyle{M}{\left({s}\right)}=$$ ”s is a math major”
$$\displaystyle{C}{\left({s}\right)}=$$ ”s is a computer science student”
$$\displaystyle{E}{\left({s}\right)}=$$ "s is an engineering student”
1)We can rewrite the given statement as: ”There is at least one student s $$\displaystyle\in{D}$$ such that s is a math major and s is an engineering student”
There is at least one” implies an existential statement, thus we need to use $$\displaystyle\exists$$ instead.
Replace ”s is a math major” by $$\displaystyle{M}{\left({s}\right)}$$, ”s is an engineering student” by $$\displaystyle{E}{\left({s}\right)}$$ and ”and” by $$\displaystyle\wedge.$$
existss $$\displaystyle\in{D}$$ such that $$\displaystyle{M}{\left({s}\right)}\wedge{E}{\left({s}\right)}$$
2) We can rewrite the given statement as: ”For every student s $$\displaystyle\in{D},$$ if s is a computer science student, then s is an engineering student.”
”For every” implies an universal statement, thus we need to use $$\displaystyle\forall$$ instead.
Replace ”s is a computer science student” by $$\displaystyle{C}{\left({s}\right)},$$ ”s is an engineering student” by $$\displaystyle{E}{\left({s}\right)}$$ and ”if-then” by $$\displaystyle\rightarrow$$. $$\displaystyle\forall{s}\in{D},{C}{\left({s}\right)}\rightarrow{E}{\left({s}\right)}$$
3) We can rewrite the given statement as: ”For every student s $$\displaystyle\in{D},$$ if s is a computer science student, then s is not an engineering student.” ”For every” implies an universal statement, thus we need to use \forall instead. Replace ”s is a computer science student” by $$\displaystyle{C}{\left({s}\right)},$$ ”s is an engineering student” by (s), "ifthen” by $$\displaystyle\rightarrow$$ and ”not” by ~ $$\displaystyle\forall{s}\in{D},{C}{\left({s}\right)}\rightarrow\sim{E}{\left({s}\right)}$$
4) We can rewrite the given statement as: ”There is at least one student s $$\displaystyle\in{D}$$ such that s is a math major and s is a computer science student”. ”There is at least one” implies an existential statement, thus we need to use \exists instead. Replace ”s is a math major” by $$\displaystyle{M}{\left({s}\right)},$$ ”s is a computer science student” by
PSKC(s) and ”and” by $$\displaystyle\wedge.$$
$$\displaystyle\exists{s}\in{D}$$ such that $$\displaystyle{M}{\left({s}\right)}\wedge{C}{\left({s}\right)}$$
5) We can rewrite the given statement as: ”There is at least one student s $$\displaystyle\in{D}$$ such that s is a computer science student and s is an engineering student, and There is at least one student $$\displaystyle{t}\in{D}$$ such that t is a computer science student and t is an engineering student” ”There is at least one” implies an existential statement, thus we need to use \exists instead.
Replace ”s is a computer science student” by $$\displaystyle{C}{\left({s}\right)},$$ ”s is an engineering student” by $$\displaystyle{E}{\left({s}\right)},$$ and” by $$\displaystyle\wedge$$ and ”not” by ~ $$\displaystyle{\left(\exists{s}\in{D}\right.}$$ such that $$\displaystyle{C}{\left({s}\right)}\wedge{E}{\left({s}\right)}{)}\wedge$$ ($$\displaystyle\exists{t}\in{D}$$ such that $$\displaystyle{C}{\left({s}\right)}\wedge\sim{E}{\left({s}\right)}$$)

### Relevant Questions Suppose a class consists of 4 students majoring in Mathematics, 3 students majoring in Chemistry and 4 students majoring in Computer Science. How many compositions are possible to form a group of 3 students if each group should consist at most 2 students majoring in Computer Science? Suppose a class consists of 5 students majoring in Computer Science, 5 students majoring in Chemistry and 3 students majoring in Mathematics. How many ways are possible to form a group of 3 students if each group should consist at most 2 students majoring in Computer Science? The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date. A gambling book recommends the following "winning strategy" for the game of roulette. It recommends that a gambler bet $1 onred. If red appears (which has probablity 18/38), then the gamblershould take her$1 profit and quit. If the gambler loses this bet (which has probablity 20/38 of occurring), she should make additional $1 bets on red on each of the next two spins of the roulette wheel and then quite. Let X denote the gambler's winnings when she quites. (a) Find P{X > 0}. (b) Are you concinved that the strategy is indeed a "winning" strategy? Explain your answer. (c) Find E[X]. asked 2020-10-23 1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are$42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is \$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance. A local school has both male and female students Each student either plays a sport or doesn't. The two-way table summarizes a random sample of 80 students.
$$\begin{array}{|c|c|c|}\hline& \text{Female} & \text{male} \\ \hline\text{No sport} & 12&15\\\hline\text{Sport}&36&17\\ \hline\end{array}$$
Let sport be the event that a randomly chosen student (from the table) plays a sport
Let female be the event that a randomly chosen student (from the table) is female. a. If the data is weight, the z-score for someone who is overweight would be
-positive
-negative
-zero
b. If the data is IQ test scores, an individual with a negative z-score would have a
-high IQ
-low IQ
-average IQ
c. If the data is time spent watching TV, an individual with a z-score of zero would
-watch very little TV
-watch a lot of TV
-watch the average amount of TV
d. If the data is annual salary in the U.S and the population is all legally employed people in the U.S., the z-scores of people who make minimum wage would be
-positive
-negative
-zero Often in buying a product at a supermarket, there is a concernabout that item being underweight. Suppose there are 20 "one-pound"packages of frozen ground turkey on display and 3 of them areunderweight. A consumer group buys five of the 20 packages atrandom. What is the probability of at least one of the five beingunderweight?
Suppose that a school has 20 classes: 16 with 25 students ineach, three with 100 students in each, and one with 300 studentsfor a total of 1000 students.
(a) What is the average class size?
(b) Select a student randomly out of the 1000 students. Letthe random variable X equal the size of the class to which thisstudent belongs and define the p.m.f. of X.
(c) Find E(X), the expected value of X. Does this answersurprise you? Worded problem: Follow these guided instructions to solve the worded problem below.
a) Assign a variable (name your variable)
b) write expression/s using your assigned variable,
d) Solve the algebraic inequality
Worded Inequality problem:
Your math test scores are 68, 78, 90 and 91. What is the lowest score you can earn on the next test and still achieve an average of at least 85? Sample Size: $$n=15$$
Regression Equation: $$y\hat{e} =0.359 - 1.264x$$
Sums of Squares :$$SSy = 35.617. SSex = 32.589, SSresid = 3.028$$
d. Determine whether the variables x and y are significant using a $$5\%$$ significance level. You may assume a simple random sample from a bivariate normal populaton.