Matthew teaches statistics at a prestigious community college. During his graduate studies, he was fortunate to be trained in SPSS, a computer software created by IBM for statistical analysis. He thinks his students could have a better chance of getting jobs in data analysis if they knew how to work with this software. He decides to have a seminar on an introduction to SPSS. Unfortunately, his college does not have the funding necessary to support SPSS. So, he reaches out to IBM directly.

After much negotiation, the company offers him access to SPSS for his seminar free of charge if two conditions are met. First, there must be at least fifty students participating in his seminar, and each student will get a limited access to the program. Second, he must advertise the seminar to his colleagues at the college. The company will provide him with twenty free full access codes to SPSS for instructors at his college, but any additional access codes must be purchased. Matthew, who already has a copy of the software for his personal use, agrees.

Matthew wants to meet IBM's conditions, but he does not want to pay for any additional license access fees. He is also limited by room size so he cannot send out mass emails to all statistics students. Therefore, he must determine how many invitations he should send out to students and faculty. Since he is new at the college, he decides to consult with one of his resourceful colleagues, Mario.

After carefully listening to Matthew's situation, Mario tells him, "I can tell you from my experience what kind of probabilities you should expect on participation. There is a 60% chance that a statistics student will show up to your seminar after receiving your invitation. Faculty members are busier, so only one out of three will show up. Given the room that you are planning to utilize for your seminar, the probability of having at least fifty students show up is at most 90%." Matthew immediately takes out his TI-84 calculator and after a couple of minutes says, "I know how many seminar invitations I should send out now."

Mario replies, "Are you trying to send out the most number of invitations possible?" Matthew responds, "Yes, of course. It is my first seminar and I want to be on the safe side." Mario agrees but asks curiously, "With the information I gave you, I see how you got the number of student invitations. But how did you work out the number for the faculty invitations?"

Matthew responds, "Well, I just made sure that the probability of at most twenty faculty attending the seminar is very high." Mario asks, "How high?" Matthew replies, "At least 95%, we statisticians like that one, don't we?" Mario nods his head in agreement and says, "Yes, I see. The problem is solved." They exchange some pleasantry and depart.

Once Matthew is out of his office, Mario calculates the probability of at least fifty students and at most twenty faculty showing up to Matthew's seminar after they receive the invitations. He thinks to himself, "Matthew is going to be fine."

Find the number of seminar invitations that Matthew will send out to statistics students and the number of invitations for faculty (or his colleagues). Determine the probability of at least fifty students and at most twenty faculty showing up to Matthew's seminar under the given conditions? Do you agree or disagree with Mario's last thought? Why