I started thinking about something recently, inspired by a problem I across in a textbook. That problem is actually talked about here: 100 Soldiers riddle
I was curious. Say you have 2 colleges, A and B. They both accept 1,000 students each. Of those 1,000 students, 900 matriculate to each institution (so each school has 900 attendees)
I was trying to think, what is the maximum number of students that could have been accepted at BOTH? And what does that tell us about your odds of getting into one college, knowing you got into the other. I have worked it out myself, and I believe the answer is 200 but I'm not sure how to prove it "rigorously".
So my solution was this. We know the maximum number of unique students is 2000. A accepts 1,000 students, B accepts another totally unique 1,000 students with 0 overlap. In this case, there's 0 students who get into both.
The minimum number of students has to be 1,800 because we know 1,800 unique students matriculated. So say A accepts students 1-800 and those students matriculate to A. Say B accepts students 1001-1800 and those students matriculate. That leaves the 200 students in the middle, that both A and B could accept, and say students 801-900 go to A and 901-1000 go to B. This "works" and proves that minimum number of unique students is 1,800.
This would imply that if you get into one, your odds of getting into the other are only 10% (200 students who got into both/2000 students who got into at least 1).
It seems obvious that the answer of 200, has something to do with 2,000 total admittances-1,800 unique matriculants = 200. But I'm not sure how I would prove that. What branch of math would this even fall under? I would think set theory but I'm not totally sure.
- Is there a more rigorous way to prove this?
- How would I generalize this beyond just 2 in a mathy way?
- Is my statement about the 10% probability correct?
