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(a) There are eight research projects with five projects about transportation and the rest are about marketing: The Research Director wishes to assign four projec...

Question

(a) There are eight research projects with five projects about transportation and the rest are about marketing: The Research Director wishes to assign four projects to a research team at random: Find the probability that the team will be assigned least two projects about marketing: marks)(6) An education specialist conducted a survey t0 investigate the type of activities participated by primary school students last summer. From the results, 80% of the students participated in STEM courses. Am

(a) There are eight research projects with five projects about transportation and the rest are about marketing: The Research Director wishes to assign four projects to a research team at random: Find the probability that the team will be assigned least two projects about marketing: marks) (6) An education specialist conducted a survey t0 investigate the type of activities participated by primary school students last summer. From the results, 80% of the students participated in STEM courses. Among those students participated in STEM courses, 75% of them participated in sports courses 10% of the students participated in neither STEM courses nor sports courses. Calculate the probability that randomly selected primary school student participated in sports courses. marks) Calculate the probability that a randomly selected primary school student participated only in STEM courses Or only in sports courses. marks) (iii) It is known that randomly selected primary school student participated in sports courses, find the probability that this student participated in STEM courses marks) The employees of a company are classified mana gerial, technical and clerical staff with percentage shares of 25%, 60% and 15%, respectively: The percentages of managerial, technical and clerical staff who are satisfied with the recent salary adjustment are 65%, 78% and 60%, respectively: If three staff are selected randomly from those who are satisfied with the recent salary adjustment, what is the probability that exactly one of the three is & technical staff? marks)



Answers

A school has 250 employees categorized by task and gender in the following table. $$\begin{array}{|l|c|c|c|}\hline & \text { Teaching } & \text { Administrative } & \text { Support } \\\hline \text { Male } & 84 & 14 & 52 \\\hline \text { Female } & 56 & 26 & 18 \\\hline\end{array}$$ An employee is randomly selected. Let $A$ be the event that he/she is an administrative staff member, T teaching staff, S support, $M$ male, and F female. a) Write down the following probabilities: $P(F), P(F \cap T), P\left(F \cup A^{\prime}\right), P\left(F^{\prime} | A\right)$ b) Which events are independent of $F$, which are mutually exclusive to $F$. Justify your choices. c) (i) Given that $90 \%$ of teachers, as well as $80 \%$ of the administrative staff and $30 \%$ of the support staff, own cars, find the probability that a staff member chosen at random owns a car. (ii) Knowing that the randomly chosen staff member owns a car, find the probability that he/she is a teacher.

Okay, So in this problem, instructor has found a way to calculate Ah students probability of passing a class based on variables of s and a where s is the students score on ah, departmental placement exam and is a number of semesters of mathematics passed in high school and a is the students mathematics s a T score. So I have that form your function working down here 0.5 times a plus six times s times end to 1/2 power. Now this problem has three parts A B and C part A wants you to find the probability that a student passes a class with we'll score of six on the placement exam four semesters of high school math and s a T score of 4 16 So they want you to find f of six for for 60. And all you do is simply plug in these about values for the variables into our function. So if we do that, we get 0.5 times a which is for 16 or 60 plus six times the square root of size and which is four. And if we were to multiply its all out, we get an answer of 52.39%. So this student in particular would have a 52.39% chance of passing a math class. Um, eso not party for TB. Ask you to find probability for another student. Another student this time with math. Score off. Uh, with the last score of four. Please don't score. Four with five years of 55 years, five semesters of high school math and a score of 300 it s a T. So all we gotta do again is play this into the formula. Pretty simple. Your points. You're a fine times 300 plus six times the square root of 20. All right, so if we were to calculate all this, this person has a 41.83 41.83 percent chance of feeling, So this person has more likely to fail the class and pass it. Now. Part C Part C asks you to find an interpret the partial prove it is with respect to end and a at the 0.45 480. So, if we were to take the partial derivative with respect and and find the value. At 45 for 80 we're gonna first need to take the partial derivative with respect to end. Let's go ahead and do that. If we took the derivative with respect to end, all were differentiating. Is, uh this terms? This is the only turn with on invariable in it, and everything else is constant. So if we were to differentiate that we get, we had to bring down the two. So we've got three times s times on to the native. 1/2 power times times s because off general. Right. So this is our part of the related with respect to when and if you plug in the 0.45 480 to arm for me long, we get two point 683683 And this is our partial derivative. Well, just to pay. No, no. They also want you to interpret what this value means. That we're gonna go ahead and do that. Type that here in this text box. Yes, text box. And it says, and this would be a great change. Read a change in the probability of passing her unit change in the lumber. Oh, it's a mess. Race a school while keeping well. Keeping reveals constant everything else. I mean, the other two variables. All right, so this is our This is our interpretation for the function, the partial derivative of the function with respect to end. Now, the other thing they want us to do is take the partial derivative with respect to A with respect to a at the point, the same point for five for 80. All right, so first, we're gonna need to take the partial derivative respect A and thankfully, we just have to differentiate this. So are partial derivative is simply yes. 0.5 So there are no variables. So our partial derivative at the 0.4 580 is just 0.5 If we were to interpret this, we're contemplative. Since we can't interpret this as well, um, this is just going to be a rate of change. Greater shame off. Read a change in a probability. Passing her unit change in the students in the students, Uh, s a T score. Well, keeping There you go. This is our interpretation for the partial derivative with just like to a and I believe that is it for this problem

Cool. Okay. Sixty. A developmental mathematics instructor at a large universe that has determined on a student's probability off success. You know, you're Estes Pass or fail, Matthew. Our general course is a function of s and an early where s is a student school auntie departmental placement exam and is a lumber up its master Sof masked man. It's past in high school and a is a student's mathematics and Santy score estimates. She asked him, Mr P, the probability off passing the course in person will be their function. And we have a range of a twenty, two hundred and hundreds as his front there to turn a frontier eight and way. Want Teo finds Feli A for students Score six hundred placement exam has taken force master of high school mass and high school and he's along grossness through. Okay, so any for no. Okay, we're just plugging numbers for A and equals four. Score six on the replacement exam, which is ass because six and s A T scores for sixty Shit. That's a if for sixty now a plaque in p equals zero point zero five. What flower for sixteen applause. That's a lie. Six times. What hundred six with one half. Now I just need to calculate that he's in our It was a dream of flying here. Yeah, close. Thanks. Twenty four a half. One for the zero ah, years What I ask you, but oh P's in person. That's right. Yeah. So he's going to be fifty two. Two point through knife, fifty two percent approximately. So relax for eight. Not be. It is similar, but different. A lumber. So this time five masters off high school mass, which is on he was five s equals four A equals three hundred. Ah, Now blocking P equals syrup on zero five. Twenty. Plus we're grew close. Six months. Fly forward tongues fine. Toothy one half. Now, if we can correct that. Ah, ze aeroplane. Fried flies three hundred a quarter Rosie hair appointment here does it for me in there. What a one point takes three percent. Not this car's nowhere probability to pass that eight three. Now. See, we want to find f sub end off this lumber and f sub bay. So we need to first find its up and And if something so f sub urn you cause differentiate this function with respect and eso, This party's become constant this part of we need to use Okay, we can write off how frequent their plans. Their five a day pull us six s water and one off into this part thereof thiss part constant. One of us is far channel for end one half on next to one House three. This is the one half. And to collect you one half we want to will have four five, seven for eighty Now, this is a This is can This is class. So we're plucking. We have three multiply forces. One half butterfly End was in Nephew five ninety one have now every Congress. That's just too right. So it caused six divided by a squared five. Um cock anus. These are spooks. Bye, squirt. Fine. Oh, which is Troop on fifth? Take Not That means we want to interpret it, right. That means one. A CE stays out of four estates. Before it was for eighty. When and increased by one. You got one more semester of mass, Angus, your person did go up by two point six. Yeah, so that's for FC on. Have sub a Yeah way. First find out someday equals it's just their appointed. There are five, right? This part is this constant way back in. That's all right. I hear this guy course they're part of their body. Doesn't even matter. So what? This means one your ass. And you was flying a five. Whenever you're a increased by along, that's your s a T score increased by one. Your probability off passing this art protest increased by three point zero five. Yeah. Yes, that's it for us.

Question eight Given? Yeah, fresh. I mean equal 40% me off. So for me, so farmers equal 25% be off juniors equal 15% be off senior equal 20%. You can use the binomial model in your mind. The probability because there are two possible outcomes. Freshman or no freshman success Student is the freshman failure. Jordan is not Fishman. The probability off success is the probability that the student is a freshman. So prosperity, equal property off freshman equal 14% question number be It's not appropriate. So modern. The, uh five trying and independence. Because the population size off 30 student is a smooth in the case off small population size, it's better to use is a hyper geometric distribution. Yeah.


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