# Reading Assignment #5 - Due 1/30/12

1. Suppose that you select a Vanderbilt undergraduate student at random. Let A be the event that the student is a junior. Let B be the event that the student is an engineering major. Describe the event AC & B.
2. Suppose the personnel manager of a manufacturing plant claims that among the 400 employees, 312 got a raise in 2008, 248 got increased benefits, 173 got both, and 43 got neither. Is this claim accurate? Why or why not?
3. Suppose A is the event that it rains today and B is the event that I brought my umbrella into work today. What is wrong with the following argument? "These events are independent because bringing an umbrella to work doesn't effect whether or not it rains today."

## 59 thoughts on “Reading Assignment #5 - Due 1/30/12”

1. 1. Event Ac&B describes a freshman, sophomore, or senior engineer.

2. This claim in incorrect. If you sum 312+248+173+43, you get 776, which is obviously more than the number of employees. The employer could have counted people how got both a raise and benefits in both the "raise" category and the "both" category. The same thing could have gone for benefits.
** However the math doesn't work out by my count so there also must have been a counting error. He must not have a Ph.D in math.
(312-173)+(248-173)+173+43=430?

3. If it raining does effect if you bring an umbrella.

4. How ridiculous was that grocery receipt explanation?

2. 1) The event (A^C)&B represents the proportion of Vanderbilt undergraduate students that are engineering majors but not juniors.
2) This claim is inaccurate. If the total number of employees is 400, then 139 employees got only a raise (312-173), 75 employees got only increased benefits (248-173), 173 got both, and 43 received neither. This adds up to 430, 30 more employees than the number of employees in the manufacturing plant.
3) Though bringing an umbrella to work doesn't affect whether it rains; rain affects whether or not you bring an umbrella to work. So the outcome of one event (whether it rains) affects the outcome of the other event (whether you bring an umbrella to work). Therefore, since the outcome of an event provides useful information about the outcome of another event, the events are not independent.
4) If the total area under a continuous probability distribution is 1, does the height of the distribution have any significance (besides showing the relative frequency of the observation, i.e. is the height standardized in any way so that one could compare 2 continuous probability distributions?)

3. 1. Ac is the complement to A and is the event that the student is a NOT a Junior so either a Freshman, Sophomore, or Senior. And B is the event that the student is a engineering major. So Ac & B is the event that the student is NOT a junior (ie. Freshman, Sophomore, or Senior) AND an engineering major.

2. No this is not accurate. According to the General Addition Rule the prob of one even occurring is P(A)+P(B)-P(A&B) where P(A&B) is the prob of both occurring. IN this Case there are only 400 workers. However after applying the general addition rule we find we have 30 too many people in the data. ((312+248)-173)+43 (neither)= 430.

3. False. These events are dependent. Even though me bringing an umbrella to work doesn't effect whether it rains, the opposite is true. Me bringing an umbrella to work DEPENDS on whether it's raining or not. The the dependence doesn't have to go both ways.

4. Why can the general addition rule be applied to both mutually exclusive, and non mutually exclusive events? While the addition rule only applies to mutually exclusive events.

4. 1) This event means picking a student who is not a junior and who is an engineering major.
2) The claim is not true because the total probability of all events together adds up to 1.94 but it must add up to 1.
3) I am not sure but I would assume that the thing wrong with this argument is that since events are independent, the outcome of one does not provide useful information about the outcome of the other.
4) I have heard that mutually exclusive events are not necessarily independent. What could be examples of that? It is a bit confusing.

5. 1. The event A^c & B refers to the complement of A and event B. In this case, the specifics would dictate that A^c would consist of freshmen, sophomores, and seniors (and graduate students, if included?), and B would necessitate engineering students.

2. Out of 400 employees, 43 did not get any change in benefits or pay, leaving 357. Out of those 357 employees, if 312 got a raise, only 45 employees exclusively received increased benefits. Subtracting this number from the 248 total employees who received increased benefits leaves 203 who, by necessity, received both a raise and increased benefits. This value disagrees with the given value.

3. Consider A as bringing an umbrella and B as a high probability of rain. In the case that A is not a cause of B, the variables can not be declared independent until the relationship is also proven nonexistent in the other direction. In this case, the high likelihood of rain (B) was likely the cause of the umbrella (A), meaning the variables were not independent.

4. Is there a formulaic way to solve for probability in a continuous distribution without a histogram?

6. 1. Suppose that you select a Vanderbilt undergraduate student at random. Let A be the event that the student is a junior. Let B be the event that the student is an engineering major. Describe the event AC & B.

The complement of A is all of the students who are not juniors. Therefore, the intersection of A's complement and B is the group comprised of all freshman, sophomore, and senior engineering majors.

Suppose the personnel manager of a manufacturing plant claims that among the 400 employees, 312 got a raise in 2008, 248 got increased benefits, 173 got both, and 43 got neither. Is this claim accurate? Why or why not?

This claim is impossible. 139 employees got raises but not increased benefits, 75 got benefits but not raises, 173 got both, and 43 got neither. The options add up to 430 employees, but the company claims only 400.

Suppose A is the event that it rains today and B is the event that I brought my umbrella into work today. What is wrong with the following argument? "These events are independent because bringing an umbrella to work doesn't effect whether or not it rains today."

Your decision to bring an umbrella is effected by the weather forecast as well as the clouds in the sky.

7. 1) The event Acompliment & B is the event that a random Vanderbilt undergraduate student is an engineering major and either a freshman, sophomore, or a senior.

2) This claim is not accurate. There are four events. According to the personnel manager’s information, 139 received a raise without increased benefits, 75 received increased benefits without a raise, 173 received both, and 43 received neither. The number of outcomes is 430, which is greater than the number of employees (400).

3) The events are independent in the way stated, but they are dependent in that whether or not it rains effects whether or not one brings an umbrella to work.

4) Is there a precise definition of a “random process”? Is there a way to measure how “random” an apparently random process is?

8. 1.) AC & B is the event that the student selected is not a junior and is also an engineering major.

2.)No the claim is inaccurate. The best way to picture this is a Venn diagram, the people that got both are in the overlapping part of the circles - 173. Therefore the number of people who only got a raise is 312-173 = 139, and the number of people who only got increased benefits = 248 - 173 = 75. The total number of people who got something is 139+75+173 = 387. Therefore people who got neither is 400 - 387 = 13 not 43.

3.) Mathematically they are independent and there is nothing wrong with the statement. In reality however, the only time people ever bring there umbrella is when it is going to rain. Therefore whether it rains or not today affects bringing an umbrella to work.

4.) Why doesn't the probability that an event will happen account for the present event itself?

9. 1) A^C & B is the probability that the selected student is not a junior plus the probability that the student is an engineering major, minus the probability that the student is an engineering major who is not a junior

2) Employees who got only a raise = 312 - 173 = 139; Employees who only got increased benefits = 248 - 173 = 75; Employees who received both = 173; Employees who received neither = 43. 139 + 75 + 173 + 43 = 430, which is above the company's employment,

3) The events may be related in the opposite direction; that is, the individual may be more likely to bring his umbrella with him if it rains.

4) None, lined up fairly well with the intro lecture that you gave on Friday as far as starting to explain those concepts!

10. 1. That event would be that you have chosen a Vanderbilt student who is not a Junior but is an engineering major.

2. This claim is not accurate. If you use a Venn diagram, 173 employees received both benefits and a raise. Since 248 got benefits overall, this means 75 employees received only benefits. Since 312 total employees got a raise, 312-173 = 139 students who received just a raise. Finally, the employer claims that 43 got neither. If you add these totals up, 75 + 173 + 139 + 43 = 430, which is more than the 400 employees the employer claims to have.

3. These two events are not independent because knowing that you brought an umbrella into works gives someone a reasonable amount of information about the other event, i.e. that it rained today. Similarly, knowing that it rained today is information that would probably suggest that you brought an umbrella to work. This means the two events aren't independent.

4. With the continuous probably section, Exercise 2.38 says that the probability of someone being exactly 180cm tall is 0 because there is no area between 180 and 180 on the graph. Since it is likely that someone is 180 cm tall in real life, doesn't this method seem erroneous?

11. 1. The event AC represents all the freshmen, sophomores, and seniors (anyone who is not a junior). And B represents all the engineering majors. The event AC & B represents the event in which the randomly selected student is not a junior and also an engineering major.
2. The claim is not accurate. Following (a slightly augmented version of) the General Addition Rule, we have 213 + 248 – 173 + 43 = 430 as the size of the sample space. According to the personnel manager, the sample space is 400. These two values are not equal. The claim is not accurate.
3. Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other. Bringing an umbrella to work provides no useful information about whether or not it rains today. However, knowing whether or not it rains today does provide information about your inclination to bring an umbrella to work. The statement is dependent in one direction while it is independent in the other.
4. Why are there an “Addition Rule” and a “General Addition Rule”? It seems kind of redundant, but I may just not be aware of the useful applications of having a distinction between a rule for mutually exclusive events and a rule that also includes non-mutually exclusive events.

12. 1. Event A^c and B is the event that the student selected was an Engineering Major that was not a Junior.
2. This claim is not true. If you subtract the 173 that got both from the two groups that got one (since the 173 are a part of each group), and then you add the numbers together, you get a total of 430 employees (which is false).
3. It is false, since whether or not it rains today influences whether or not you bring your umbrella, making the two event not independent
4. I am confused on just exactly how the bins of histograms and things such as medians and quartiles work. If the median is halfway into the bin, is the median the bin or the number? Same with the quartiles.

13. 1. The student is not a junior and is an engineering major

2. No. Adding up the total number of people does not equal 400. 139(only raise) + 75(only benefits) + 173(both) + 43(neither) = 430, which is not equal to 400.

3. The events are dependent because a person would probably not bring their umbrella to work unless there was a decent change that it would rain that day.

4. For the law of large numbers, what value of n is usually considered large enough so that the data converges?

14. 1) Ac&B describes the situation that the student is not a junior (is a freshman, sophomore, or senior) and is an engineering major.

2) 400-173 = 227 employees who go either a raise, increased benefits, or neither. 312-173 = 139 who got only an increased raise and 248-173 = 75 who got only increased benefits. To find the number of employees who got neither a raise or benefit, 227 - 75 - 139 = 13 != 43 which implies something about the manager's claim is inaccurate.

3) These arguments are not independent because if it's raining you are more likely to have an umbrella. The above argument only takes into account that B does not impact A, but not the fact that A impacts B.

4) What is the complement of an infinitely large distribution?

15. 1) The event Acompliment and B means that the selected student must be an engineering major and not a junior.

2) The claim is not accurate. The number claimed is more than the number of sample space.

3) The argument is wrong because the event is not independent if the outcome of event A is known, then the outcome of event B can be determined.

4)In exercise 2.38, why does the probability of exact 180 cm is 0? In a large and random enough sample size, there can exist sample with exact height of 180 cm.

16. 1) It is the probability of the student being an engineering major and not a junior, so either a freshmen, sophomore, or senior.
2)No, because if you add all of the probabilities of getting each thing, you have to subtract the probability of getting both though. You end up with 1.075, and since it is greater than 1 this claim is not accurate.
3) This argument is flawed because it doesn't take into account the opposite argument. While bringing an umbrella to work doesn't affect the weather, the probability that it will rain does affect the probability that you will bring an umbrella to work. Therefore, these events are not independent.
4) Is there ever a point where continuous distributions are no longer as accurate as histograms with bigger bin sizes, since a continuous distribution would require very accurate data and sometimes data collection results in errors in data?

17. 1) The event Ac & B is all students that are engineers that are NOT juniors.
2) The claim COULD be accurate, but if the manager was lying then it is not, but in terms of the numbers being able to work out, yes, it is accurate.
3) because whether or not it rains has an effect on whether you bring an umbrella to work each day. Unless you are a super careful person/are very pale and need it to shield yourself from the sun.
4) can you explain a little bit more exactly what independence means? Can there be no link whatsoever, or just no link that would actually effect the two?

18. 1. The event Ac & B means that the student that you chose at random is not a junior but is an engineering major.

2. This claim is not accurate because if 312 employees got one benefit and 248 got another, then there were 560 benefits given out. Then if 173 employees got both benefits, 387 total employees received benefits. (560-173). Finally, if 43 employees received neither benefit, then the total number of employees according to the manager was 430, which is not the actual number of employees.

3. This statement is incorrect because for two events to be independent, neither of them can affect the other. Bringing an umbrella to work does not affect the chance of rain, but the chance of rain does affect whether or not an umbrella is brought to work.

4. Why is the chart with the negative money spent on milk included?

19. 1) the event is one that the student is either a freshman, sophomore, or senior engineer major.
2) this claim is inaccurate because (312 - 173) + (248 - 173) + 173 = 139 + 75 + 173 = 387. This number represents the total of people that either got both an increase in benefits and a raise, people who just got a raise, and people who just got an increase in benefits. This number would conclude that only 13 people did not receive either, not 43 people.
3) These two events are not independent because whether it rains provides useful information on the decision whether to bring an umbrella to work.
4) the concept of continuous distribution really confuses me.

20. 1. Ac is the complement of A and includes all the possible outcomes not already included in A, in this case, all undergraduate Vanderbilt students that are not juniors. B is the event that the student is an engineering major, which is independent of whether or not the student is a junior.
2. This claim is accurate because for each scenario, the number of employees that got either a raise or increased benefits overlaps each other. In better words, the 248 employees that got increased benefits were not an additional 248 people from the 312 that got a raise. Of the 312 that got a raise, a number of them also got the increased benefits. Same for each of the other categories in this situation.
3. It is not true that these events are independent. This is because when two processes are independent, the outcome of one provides no useful information about the other occurring. If A occurred meaning that it rained today, this would make B likely to occur as a result of A, bringing an umbrella to work. B is therefore dependent of A but A is independent of B.
4. How would you be able to categorize a specific sample space for lets say, an entire population?

21. 1. Suppose that you select a Vanderbilt undergraduate student at random. Let A be the event that the student is a junior. Let B be the event that the student is an engineering major. Describe the event AC & B.

The student is a junior engineering major.

2. Suppose the personnel manager of a manufacturing plant claims that among the 400 employees, 312 got a raise in 2008, 248 got increased benefits, 173 got both, and 43 got neither. Is this claim accurate? Why or why not?

No, this is not correct. 312+248-173+43=430, which is too many employees.

3. Suppose A is the event that it rains today and B is the event that I brought my umbrella into work today. What is wrong with the following argument? "These events are independent because bringing an umbrella to work doesn't effect whether or not it rains today."

The rain affects whether or not you bring your umbrella to work, so they are still not independent.

How do you do calculations for dependent events?

22. 1. The probability of the event AC occurring is 1 minus the probability of event A occurring, or P(AC) =1-P(A). This is because AC is the compliment of A. This means that AC is the event that the student is not a junior. AC & B is the probability that the event that the student chosen at random is not a junior and an engineering student. To get this probability we can multiply the probability of the student being an engineer with the probability that the student is not a junior. Or P(AC)*P(B)=P(AC &B).
2. This is not accurate because according to the data given there is 430 employees who work at the plant, when in fact there is only 400 employees. If we add the number of people who got a raise with the number of people who receive benefits, we get 560. But then we must subtract the number of people who are counted twice, which are the people who received both a raise and benefits. Then we would get 387 employees. After that number is calculated, we must then add the employees who got neither a raise nor benefits. The final number would be 430 employees, which contradicts the first claim that there are only 400 employees.
3. The argument states that the event of raining has no influence on the event of bringing an umbrella to class because bringing an umbrella to class has no effect on the weather. This statement is false because if we take the inverse of this statement, we can see that the event of rain and the event of bringing an umbrella influence one another. If it is raining, then it is more likely that an umbrella will be brought to work. From this statement, we can see that they do influence one another and that the two events are not independent.
4. It seems that independent variables are ones in which the outcome of one event does not affect the outcome of another. This seems mostly determined by logic and analyzing the situation with no clear data to reference. My question is, what if two events are connected but in a very convoluted way? Such as if a butterfly flaps its wings in America and a tornado happens in japan.

23. 1. The student is an engineering major that is not a junior.

2. 312+248-173+43=430 employees. This is more than the total number of employees, so the claim is inaccurate.

3. While A may not cause B, B could still cause A making the variables not independent. In this example, a person may be more likely to take an umbrella with them if it is already raining or if there is rain in the forecast. (Of course, I almost never use umbrellas because I prefer a rain jacket. If someone never used an umbrella, then the two events would be independent.)

The special topic on continuous distributions was rather brief and seemed to treat a 'continuous' distribution as nothing more than a distribution with very fine bin sizes. Is this how continuous distributions are normally treated rather than more of a pure mathematical sense of continuous?

24. 1.
The event would include freshmen, sophomores and seniors and all engineering majors (does not include juniors who are not engineering majors).
2.
This claim is not accurate. If 173 people got both, then that still leaves 139 people who got only a raise and 75 people who only got benefits. Add that to the 43 people who got neither and you have 430 total employees.
3.
Two events are only independent if the outcome of one provides no useful information about the other. Knowing that it rains is useful information because you are more likely to bring an umbrella if it does, therefore the two events are not independent.
4.
I'm confused about Figure 2.8, the probability distribution of household income. Why use discrete probability numbers when measuring this percentage? Shouldn't it just be a percentage and not a probability?

25. 1. Since the complement of A is the event that the student is not a junior. A^c & B is the event that an undergraduate is not a junior (freshman, sophomore, senior, 5th year senior, etc.) and is an engineering student.
2. No, it is not accurate. Out of the 400 employees, if 312 got a raise, thats a maximum of 86 that got neither. Then subtract 173 from the 248 that got increased benefits since those 173 already got a raise, add that number to 312, and there are 387 employees that got one or the other. That means 13 employees got neither.
3. If the weather forecast says there is a chance it rains, then you will be more likely to bring your umbrella into work. Therefore, the events are not independent.
4. If two events were very slightly dependent on one another, would you say they are partially independent or partially dependent or something else?

26. 1.
Given event A describes a junior student, and event B describes an engineering major, A comp & B describes undergraduate Vanderbilt students majoring in engineering who are NOT juniors.

2.
This claim is not accurate. Given the data is accurate, removing redundant counting gives a total workforce of 430 employees, which is more than the listed 400. (312 raise -173 both)+(248 benefits -173 both)+43 neither+173 both = 430 employees.

3.
That statement is incorrect, because independency must be evaluated from all relevant sides, and so if the situation is reevaluated it might become evident that the rain may in fact bring about the umbrella, rather than the other way around.

What is the mathematical model for determining independencies?

27. 1) The event described is that the student is not a junior and is an engineering major.

2) The claim cannot be accurate. We know that the number of people who got just a raise is 139 (312-173), the number of people who got just increased benefits is 75 (248-173), and the number of people who got neither is 43. This adds up to 257, less than the 400 hundred employees at the plant.

3) The events A and B are not independent because whether it rains or not does affect whether you bring your umbrella to work.

4) Is the only way to prove A and B are independent is if P (A and B) = P(A) ∗P(B)?

28. 1) Suppose that you select a Vanderbilt undergraduate student at random. Let A be the event that the student is a junior. Let B be the event that the student is an engineering major. Describe the event AC & B.
> The set A-compliment and B contains all engineering majors that are not juniors.

2) Suppose the personnel manager of a manufacturing plant claims that among the 400 employees, 312 got a raise in 2008, 248 got increased benefits, 173 got both, and 43 got neither. Is this claim accurate? Why or why not?
> This claim is not accurate. You can not construct a set with these properties. You can construct a set with all the properties except the neither group.

3) Suppose A is the event that it rains today and B is the event that I brought my umbrella into work today. What is wrong with the following argument? "These events are independent because bringing an umbrella to work doesn't effect whether or not it rains today."
> It has to be shown to be independent in both directions. While the stated claim is true, whether or not it rains today does effect bringing an umbrella.

> None

29. 1. The event A^c & B would be the event that the student is a freshman, sophomore, or junior and is also an engineering major.
2. No this claim is not accurate. If you total the amount of employees who either got a raise or increased benefits and then subtract the 173 who got both this leads to 387. Then by adding in those who got neither you get a number larger than 400 showing this claim cannot be accurate.
3. Two independent events do not necessarily have anything to do with "effecting" the other but instead are independent of each either because knowing the outcome of one provides no information about the outcome of the other. While the events in this statement may be independent the word "effect" makes this argument incorrect.
4. I was a little confused by some of the exercises dealing with independence.

30. 1. A^c is the event that the Vanderbilt undergraduate student is not a junior, and B is the event that the Vanderbilt undergraduate student is an engineering major. So the event A^c AND B is the event that the Vanderbilt undergraduate student is both not a junior and is an engineering major.
2. This claim is not accurate because when the numbers are compared, 139 employees got a raise, 75 employees got increased benefits, 173 got both, and 43 got neither. These values total up to 430, which is more employees than exist at the plant.
3. The problem with the statement is that independence doesn't mean that they do not affect one another, but that one event does not provide any information about the outcome of the other.
4. Independence seems tricky.

31. 1. This is the event that a random Vanderbilt student is not a junior and is an engineering major.

2. This claim cannot be accurate because according to the numbers, there were a total of 430 employees, not 400.

3. Independence exists when either one of the outcomes has an effect on the other. In this case the fact that it's raining affects whether or not you bring an umbrella to work so the events are not independent.

4. On a continuous distribution what does the y-axis value represent for a given x-axis value? Is a continuous distribution only useful when talking about a range of values?

32. 1. P(Ac&B)=(1-P(A))+P(B)
2. No because 312+248-173+43 != 400
3. it is wrong because the probability of you bringing your umbrella is effected by whether or not it rains.
4. none

33. 1) The probability that the student is a junior AND an engineering major.
2) No, this is not accurate. There must be exactly 387 people who got AT LEAST one type of perk (312+248-173), which leaves room for only 13 who did not (400-387).
3) It only gives one half of the picture. Bringing an umbrella may not effect the rain, but raining effects bringing the umbrella.
4) I only have a snarky question: what did the grocery list have to do with probability? Was it actually there for a reason, other than to propose that we should all buy milk to save money?

34. 1) The Event A^c & B indicates the probability that the student is NOT a junior, but IS an engineering major.
2) No. (312 - 173) + (248 - 173) + 173 + 43 = 430. There is a math error somewhere in the manager's calculations.
3) These events are not independent. There is an increased probability that I will take my umbrella with the knowledge that it is raining.
4) A better explanation of the notation for events, probability, etc. Just a basic chart would be helpful.

35. Given that A means the Vanderbilt student is a junior, and that B means that the Vanderbilt student is an engineering major, A(c) & B means that we have the intersection of the complement of subset A and the subset B, of the larger sample space S, which represents all Vanderbilt undergraduates. A(c) means that the undergraduate student is *not* a junior (and is therefore freshmen [first-year students, as they like to be called, fresh meat, as they are actually called], sophomores, seniors, and super-seniors), and B is all the engineering students. Therefore the intersection is all those freshmen, sophomores, seniors, and super-seniors who are all studying engineering.

Assuming that the overlap groups (those who got both added bonuses) report as people who gained each individual bonus as well, this means that each person in the overlap group has reported 3 times. So if you added up all the numbers and subtracted 2 times the overlap group, it should add up to 400. 312+248+173+43-173-173 = 430. Therefore the claim is not accurate.

According to the book, two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other. While bringing an umbrella does not affect the weather, the weather affects bringing the umbrella. Knowing if the weather forecast was for rain or sunshine would certainly affect your umbrella carrying decision one way or the other.

How do you apply a continuous distribution to discrete data?

36. 1. This event describes the event the student is not a junior and an engineering student, simply: non-junior engineers.
2. No it is not accurate. From both of individual categories, it is necessary to remove 173 to account for those who just received that perk in 2008 (ie. just received a raise or just received benefits). The total is then 257 people, which adding the additional 43 people who received none adds up to 300 people, which is 100 less than expected. All employees should have fit into one of these categories, making this story false.
3. It is not correct because it is not correct, like how people are more likely to bring an umbrella with them if it will rain. Although the weather does not wait until people bring umbrellas until it rains, people (normally) wait to bring umbrellas if it will rain. They are not necessarily independent.
4. Getting at more complex studies (involving human activity), where is a general cut off for saying something is random? Ie. for a psychological study in the Nashville are that was thought to be random, how do you mathematically know that the local culture/diet does not affect the study?

37. 1. Students who are not juniors major in engineering in Vanderbilt.
2. 312 + 248-173 +43=430 does not equal to the total number of 400 employees. This claim is not accurate.
3. If it rains today whether or not does effect bringing an umbrella to work.

38. 1. Freshmen, Sophomores and Seniores who are engineering majors.

2. No because the 173 employees who got both are already accounted for in the 312 and 248.

3. B is dependent on A.

4. Not too complicated. I might need to practice when to use add probabilities and when to multiply.

39. 1. The event would be describing the probability that the student is not a junior and is an engineer. This includes all freshman engineers, sophomore engineers, and sernior engineers.

2. No because the total of the numbers equals 430 instead of 400. All 400 people got benefits, a raise or neither.

P(benefits or raise) = P(benefits)+P(raise)-P(both) = 248 + 312 -173 = 387

We then must include those who got neither

P(neither) + P(from above) = 43 + 387 = 430!!

3. The problem with the statement is that is not what independent means. Independent means that knowing one does not help you to figure out the other. The two statements are independent because if he brought the umbrella to work you still have no idea whether it will rain or not.

4. What is the error caused by using the distribution curve to calculate probabilities rather than using the exact histogram?

40. 1. Suppose that you select a Vanderbilt undergraduate student at random. Let A be the event that the student is a junior. Let B be the event that the student is an engineering major. Describe the event AC & B.
This event describes a student who is not a junior and is an engineering major.

2. Suppose the personnel manager of a manufacturing plant claims that among the 400 employees, 312 got a raise in 2008, 248 got increased benefits, 173 got both, and 43 got neither. Is this claim accurate? Why or why not?
The counts of all disjoint events are as follows:
got raise, but not benefits: 139 (312-173)
got benefits, not raise: 75 (248-173)
got both: 173
got nothing: 43

Summing all disjoint events totals to 430 employees, not 400. Therefore, this claim is NOT accurate.

3. Suppose A is the event that it rains today and B is the event that I brought my umbrella into work today. What is wrong with the following argument? "These events are independent because bringing an umbrella to work doesn't effect whether or not it rains today."
Independence requires NEITHER to affect the other. Thus A cannot affect B and B cannot affect A for A and B to be independent. This statement states that B does not affect A, BUT DOES NOT state that A does not affect B. In fact, its obvious that A does affect B, since if it rains more people will bring an umbrella. Thus, the two are not independent because the statement A does not affect B is not satisfied.

The reading was straightforward. No questions here.

41. 1. The student is not a junior and is an engineering major
2. No, because employees with raise + employees with benefits - employees with both + employees with neither does not equal the total number of employees
3. Rain might affect whether or not you bring your umbrella, which means the events are dependent.
4. It seems like a situation could arise in which two events are not independent, and yet when you multiply their probabilities you happen (by mathematical coincidence) to get the probability of both events happening. Is this possible?

42. 1. This event refers to the freshmen, sophomores and seniors who are engineering majors.
2. Let A be the event where people got a raise in 2008 and B the event where people get increased benefits. (A & B) would then refer to people who receive both a raise and increased benefits, while the compliment of (A U B) are for people who receive none. When we construct the Venn diagram, we see the total sums to 430, which is 30 people more than the number of employees. Therefore, this claim is not accurate.
3. Therefore, although bringing one won't necessary make it rain, a person will most likely have an umbrella when it is known there is a chance of rain. There might be a small dependence between the two to negate their independence of each other.
4. In constructing table 2.7 on page 61, why isn't the probability of dice sum be 1/36 across all dice sums?

43. 1. The event A^C & B describes the event that a randomly selected Vanderbilt undergraduate student is a freshman, sophomore, or senior engineering major.
2. The claim is not accurate, because if 173 got both, then 312 - 173 = 139 employees got only a raise, and 248 - 173 = 75 employees got only increased benefits. With 43 who got neither, that makes the total number of employees 139 + 75 + 43 + 173 = 430 instead of 400.
3. The argument only considers one of the possible relationships between events A and B. It does not, therefore, consider the fact that the one is more likely to bring their umbrella to work when it is raining, which makes the events dependent.
4. I didn't have any questions about the reading, because I have encountered all of the material in previous classes.

44. 1.) This event is interpreted as "a Vanderbilt student who is not a junior and also an engineering major".

2.) The claim is not accurate. 312-173 = 139 is the number of people who got a raise only. 248-173 = 75 is the number of people who got increased benefits only. So 139+173+75 = 387 is the total number of people who got one or the other or both. This number should also be equal to the total number of people minus the number of people who got no benefits at all. However, this is not the case: 400-43 = 357. The numbers do not match up and so the claim is false.

3.) Although bringing your umbrella to work in no way affects the probability of rain, the probability of rain affects whether or not you bring your umbrella into work. Hence they are not independent. As the probability of rain increases, so does the probability of bringing your umbrella to work.

4.) Is probability something that exists independently of our minds, or is it simply compensation for our lack of knowledge of an event? That is, do we use probabalistic models for some systems because deterministic models for such systems are difficult to find or would be computationally intractable, or because those systems are somehow intrinsically probabalistic?

45. A)

It means that every student who is not junoir and has an enfineering major !!

B)
For this question, simple caclculation is needed.

1) We have 400 employees.
2) 173 of them took benefits and rise ==> 400 -- 173 = 227 employees left.
3) Also, take of 173 from 312 (rise) and another 173 from 248 (beneits)
===> 312 - 173 = 139 (who only took rise) .
===> 248 - 173 = 75 (who only took benefits).
4) suppose to be that : # of only benefits + # of only rise + # of took nothing = 227
However, 139 + 75 + 43 = 257 which is not equal to 227 .
5) the personnel manager of a manufacturing is liar !!

C)
It seems that they are not totally independent from each other. It is true that rain is independent from the umbrella (the umbrella does not control the rain!!). However, the rain control the umbrella. For example, if it is raining, the chance that I am going to take the umbrella is much higher than if it is not raining. In other words, it is one way dependency.

D)
The third question was little bit confusing

46. The student is not a junior but is an engineering major.

Suppose the personnel manager of a manufacturing plant claims that among the 400 employees, 312 got a raise in 2008, 248 got increased benefits, 173 got both, and 43 got neither. Is this claim accurate? Why or why not?

It is incorrect, because we have 357 people who got some form of raise or benefits, and out of those only 124 who got only one. The "both" category covers all but 139 of the raises and 74 of the increased benefits, meaning there must be an additional 212 people not in the "both" or "neither" category, but as we showed above there are only 124. Therefore, the statement must be false.

You only looked at the argument one way. Sure whether it rains isn't dependent on the tiny being who is you, but what you do might be dependent on the wether.

Is there a general rule of thumb for when to use histograms and when to use continuous graphs or are they both completely interchangeable with one another?

47. 1. That will be the set of all freshmen, sophomore and senior with engineering major.
2. Wrong. According to the data, 139 got raise only, 75 got benefit only, 173 got both, and 43 got neither. Then total number would be 139+75+173+43=430 which is not equal to 400.
3. The statement after "because" is right but that doesn't result in the conclusion. Actually, whether I bring an umbrella depends on the weather today.
4. I don't have a question for this reading.

48. 1. The event is an engineering student that’s not a junior
2. Because the total number of employees in this case using the general sddition rule is (312+ 248-173)+43=430 which is greater than the actual number of employees which is 400.
3. The argument is wrong because the fact that it rains affects the decision of whether or not to carry an umbrella to work so they are not independent events.
4. How do you better distinguish continuous distributions?

49. 1. P(Ac or B)=P(Ac)+P(B)-P(Ac and B) or probablility of selecting a freshmen, sophomore or senior engineer.

2. No, only 23 could have gotten neither. 312+248-173=23

3. Normally one only brings their umbrella on a day with a significant chance of rain, thus the outcome of the first event significantly influences the second one.

4. When we have 3 events in a vinn diagram how do we get out the overlaping double pieces?

50. 1. The student is an engineering major and he/she is not a junior
2. It's not accurate. According to the calculation, 312+248-173+43 = 430 > 400
3.I think that it is wrong because I can watch the weather forecast and determine whether I should bring the umbrella. Therefore, if there is a high probability of raining, I will bring the umbrella. So they are dependent.
4. Whether the events of the weather of two cities in the same state are dependent or not?

51. 1. The probability that the student chosen is not a junior and is an engineer.

2. No. If 43 people got nothing, then 357 people would have had tog et something which is not the case.

3. The other way is true though. Raining does affect whether you bring an umbrella to work or not.

52. 1. the probability that the student is an engineering major that's not a junior.

2. no because the amount of people that got a raise and increased benefits would be over 400. If getting a raise is included in increased benefits, then there should be more people that got increased benefits than the ones that received raises.

3. the two are not independent because knowing whether it rains or not can affect the chance of having an umbrella.

53. 1. This event is the complement of A and B, so it is all students who aren't juniors who are engineering students. In other words, it is all freshmen, sophomore, and senior engineering majors.
2. This claim is not accurate. If imagining the Venn diagram of who got a raise and who got increased benefits, with the center being those who got both, there are (312+248)-173 inside of the entire diagram, which is a total of 387. This only leaves 13 left to have received neither, but the problem claimed that 43 got neither, thus, it is inaccurate.
3. This argument doesn't hold water (no pun intended) because the fact that you brought your umbrella to work today was most likely caused by the fact that it was raining outside, or that there was rain in the forecast for today.
4. Was there any point to the grocery receipt lists on p. 61? I was very confused as to where that was going, and then never really saw it resolve.

54. 1) This event is selecting all of the undergraduate, engineering major students who are not juniors (freshman, sophomores, and seniors).

2) This claim is inaccurate. If 43 employees did not receive anything 357 employees received at least a bonus or benefits. Out of this amount 312 people received a raise showing that 45 people did not. 248 received benefits so 109 did not (of the 357). Of 357, 154 people did not receive either a bonus or benefit. This added to the 43 people who received nothing gives 197. This leaves 203 people who received both which is greater than the given 173. I think...

3) These events can be unrelated or independent. Many bring umbrellas out thinking it is going to rain when it turns out to be a sunny day. You could have been storing that umbrella for a later day. One doesn't necessarily cause the other. Some people may use rain jackets instead of umbrellas.

4) Is there a mathematical formula so solve questions like #2 on this reading or do you just observe and add the numbers?

55. 1. The event of freshmen, sophomores and seniors who are engineering majors.
2. Not accurate. 400-[(312-173)+(248-173)+173]=13 which is not 43.
3. The events are NOT independent because whether or not it rains today WILL effect bringing an umbrella.
4. none.

56. 1. This means that he is not a junior (i.e he is either a senior, sophomore, or freshman) and he is an Engineering major

2. If 173 got both, then 139 (312-173) got only a raise, and 75 (248-173) got only benefits. 139+75+173 = 387. Thus, there should be 13 people with neither, not 43. The statement is not accurate.

3. The statement is wrong because the converse is true. Assuming rationality, whether or not it rains would affect whether or not your bring an umbrella to work. This makes the two events related.

4. If, according to the text, probability is the proportion of times an outcome would occur if we observed a random process an infinite number of times, doesn't that disprove the conjunction fallacy? Thus if one has rolled a die 1 million times and has got only heads, doesn't that mean that for the probability of heads or tails to be 1/2, at some point he is going to have to get only tails for the same amount of rolls?

57. 1. Ac represents every other outcome possible and is the complement of A, meaning the student chosen would be a freshman, sophomore, or senior. B represents another independent outcome, of which the probability can be jointly calculated with A using the product rule.

2. No, the two options are not independent and therefore the overlap causes the individual values to be too high.

3. The events are not disjoint because they share an outcome -- if it rains, then you will bring your umbrella into work and vice versa.

4. How do you prove the product rule for independent processes?

58. 1. The event A^c & B will include all freshmen, sophomore, and senior engineering students. These are all of the complements that are not included in A.
2. Not this claim is not accurate. When all of the variables are added together they equal a sample size of 430 which is larger than the original sample size of 400.
3. The events are not independent of each other because the event that I brought my umbrella into work today is completely dependent on the event that is supposed to rain today. So the two events are dependent of each other.
4. I need more help with the idea of probability density function.

59. 1) The compliment of A and B would represent all non-juniors who are engineering majors.
2) No, if you subtract our the 173 people who received "both" from the pools of people who received raises and benefits, the sum of all three categories comes to 387, which is more than the reported number of (400-43) = 357 who should have gotten one or the other, or both.
3) Because B can effect A. If it does rain, there is a high probability that you will bring your umbrella to work today.
4) I thought most of the chapter made sense, since it was mostly vocabulary definitions. I really wish they would stop using the chart of different sized cars all the way back from Chapter 1 though hehe.