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Here is the solution I presented:

(a) P(G|B) = (5/24) * (.125) = .026. Since you select ONE senior manager, then P(B) is 4/23. Now, if we are dealing with a second employee who is also a senior manager, then (4/23) * (.125) = .022. So, these are the criteria. Therefore, the TOTAL probability under these conditions becomes .026 * .022 = .056%

(b) Let's deal with the partner now. (6/24) * (.375) = .094. Since you select one employee who is a junior partner, then (3/23) * (.500) = .065. Taking the data together, then: (.094) * (.065) = .611%

(c) (10/24) * (.125) = .052. If you choose one senior level employee, you can use it to find the probability you are looking for in this question.

(9/23) * (.500) = .196. Then 8/22 * .500 = .182. So, the total probability has to be .19%

The student is stuck and I am not sure where I am failing to be more clear: The student asks "Why do you calculate P(G|B)? I know that it is the probability that an employee is senior-level given that they are a manager. Why do you calculate this?"

Student subsequently asks:

Shouldn't $P(G|B) =\frac{P(GandB)}{P(B)}$ ? I think you meant $P(G and B) = \frac{5}{24} * 0.125$ Also you are assuming that the $P(G)$ is constant? Initially there are $0.125*24 = 3$ seniors After the first senior manager has been chosen, the $P(G)$ changes to $2/23$ right? Or am I misunderstanding something?

I explained by getting into a discussion about the intersection of events. I also clarified it in a different way:

You have 24 total employees. 10/24 are associates, 6/24 are partners, 5/24 are managers, and 3/24 are entrepreneurs.

They can either be entry, junior or senior according to your question. Also, you provide the data: your question says that entry level is 50%, junior 37.5% and senior 12.5%.

Think of it then this way:

50% you can write it out as 1/2. So 1/2 * 10/24 = 5/24 ----> entry level associate.

3/8 * 10/24 = 5/32 > there's your junior level associate.

1/8 * 10/24 = 5/96 > there's your senior level associate.

Now look again at your question part A. You can break it down with fractions:

P(E) = 5/192 + 5/192 =10/192 = 0.0520

I can go on, but can anyone let me know their thoughts? I am unsure how to get this across to the student and I've already tried 3 ways, but he is still stuck.

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    $\begingroup$ Random initial thought on a quick first reading of the question: In (a), is "two of the people" to be interpreted as "exactly two of the people" or "at least two of the people"? Similarly for (b) and (c). $\endgroup$ Jun 7 at 14:46
  • $\begingroup$ I did not write the question, but it's a good point. I would say exactly two of the people. $\endgroup$
    – Wasp
    Jun 7 at 15:20
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    $\begingroup$ I don't know what the student was thinking, but just reading the problem confused me. There's no information about how an employee's classification and level are correlated. You seem to assume they're independent, but that's impossible unless levels get assigned to fractions of employees. $\endgroup$ Jun 7 at 18:10
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    $\begingroup$ This problem is completely unsolvable with the given information. We have no information on how the employees being met with are chosen (are we to assume they are chosen uniformly at random?) nor what the relationship between seniority and position type is. $\endgroup$ Jun 7 at 19:09
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    $\begingroup$ It's not the first time I see this type of a problem, and I am completely at loss. Clearly the author of the problem does not have even faintest understanding of probability and statistics. How come they've wrote a textbook? How come it was printed, wasn't it peer-reviewed? How come anyone actually commissioned this textbook and subjected their students to it? If we have such a high drop-out rate in Academia, why shortage of people who are able to write a decent elementary textbook? $\endgroup$
    – Kostya_I
    Jun 10 at 10:05
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I see a few things going on here that may be confusing your student.

  • The problem itself is a problematic one; it assumes but does not state that seniority is independent of position, and that the employee of Company II chooses randomly who to meet with. The first assumption might be considered realistic, but the second definitely is not! As a general rule, students get very confused when presented with a problem that's asking them to make unreasonable assumptions.
  • Worse, the problem presents a situation which cannot happen. If we are to believe that seniority is independent of position, that 1/8 of the employees are senior, and that five employees are managers, then there is only 5/8 of a senior manager! That means that the correct answer to problem (a) should be zero, because you don't have two entire senior managers to select.
  • In your solution: What are the events "G" and "B"? From context, using the numbers you're giving, it looks like "B" is "a person is a manager" and "G" is "a person is senior". But which person are you talking about, since four are being selected? And why these variable names? This is contributing to some of your student's confusion; when they say P(G) should go from 5/24 to 2/23, that's because they're misinterpreting the "5/24" as the probability of being senior, and then calculating "the number of seniors remaining divided by the number of people remaining". The best way I know to teach using this kind of notation is to be extremely explicit about what each event is, and to have the variable used in some way connected to the word; for example, maybe "$S_1$" would be "the first person chosen is senior" and "$M_1$" would be "the first person chosen is a manager".
  • Using conditional probability here is an interesting choice. If "B" does mean "a person is a manager" and "G" means "a person is senior", then "P(G|B)" would mean "the probability that a person is senior, given that they are a manager" or in more ordinary English "the probability that a manager selected at random is senior", which here would be 1/8. Your student is correct to say that the calculation you're doing would be more accurately notated as "P(G and B)", because "and" is the connective that generally represents intersectional probability.
  • Finally - your solution has an error! Your calculation in (a), for example, gives the probability that two employees selected in a row are both senior managers (discarding for the moment the inconsistencies in the problem noted above). But that isn't what the question is asking; four employees are selected, and only two must be senior managers.

I would write out a recommended solution here, to try showing to your student, but after a few minutes of thinking I'm forced to realize that the issues with the problem that I noted earlier make that impossible. If we try the careful book-keeping you were doing, keeping track of how many managers are left and so on, then we rapidly run into the "5/8-of-a-manager" issue; if we don't, then your student's complaint about assuming a probability is constant when it isn't comes up. So, my recommendation would be to walk through the problem with the student and point out what's wrong with it. A student can benefit a lot from seeing how a problem can be "wrong"!

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    $\begingroup$ Great response, thank you. As for your questions, they come from the labels I shared with the student initially. Thus, P(A) = 10/24. Managers? P(B) = 5/24, Partner: P(C) = 6/24, Entrepreneur: P(D) = 3/24, Entry-level: P(E) = .500, Junior-level: P(F) = .375 Senior-level: P(G) = .125 $\endgroup$
    – Wasp
    Jun 7 at 21:48
  • $\begingroup$ Ah yes! Thanks for pointing that out with regard to the calculation in (a)! $\endgroup$
    – Wasp
    Jun 7 at 21:50

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