I made up a counting/probability problem and I'm having trouble thinking about it the right way

countingishard

New Member
Some background: I am a pure mathematician but have always struggled with combinatorics and never took a formal probability course. I am finally motivated to learn the material, but I'm still having trouble abstracting the problems appropriately so that I can solve them.

That being said, I've been working through the examples (and eventually the exercises) in Mathematical Statistics: A Unified Introduction, where the author gave the following example (I'm going to paraphrase): Your classmate has been assigned the most popular teacher in the grade for the last 4 years. As there are 5 teachers per grade, you deduce that there is a 1/5^4 = 0.0016 chance of this occurring, making you believe that more than luck was involved.

I found this to be very misleading, because while there is a 1/625 chance that she in particular got that assignment, that doesn't mean it's particularly rare that some classmate get the best assignment 4 years in a row. So I generalized the problem as follows:

Suppose that there are 100 students in your year and 5 teachers per grade (students divided evenly among them). What is the probability that someone in your grade got the best teacher 4 years in a row?

I decided to then rephrase the problem as follows: Suppose there's a bag of marbles consisting of 20 black marbles and 80 white marbles. You distribute the marbles among 100 people 4 times. What's the probability that someone gets 4 black marbles in a row? Equivalently, what's the probability that no one got 4 black marbles in a row? (I'm not sure which if these is easier to count.)

I've been working on this for the past couple of days and the deeper I get and the more small scale examples I try, the more I feel like I'm missing the right way to think about it. At this point, I'm not even sure my rephrasing of the problem is correct, but I find it equally difficult to solve. I think I can solve it if it's just two rounds of distributing marbles rather than four. For each of the 100 choose 20 ways the marbles can be passed out, there are 80 choose 20 ways (number of ways to distribute the black marbles among the spots that were taken up by white marbles previously) that the marbles can be distributed so that no one who got a black marble the first time gets a black marble the second time. But then what?

Any help/guidance is appreciated, I just need to stop driving myself crazy with this problem.