# Tough probability problem (any takers?)

1. Dec 29, 2009

### hoodleehoo

I'd like to think I know a little about math and some basic probability based on the class I took in college, but then again maybe I don't. lol Anyway, I have no idea how to figure the probability on this. Anyone care to take a crack at it for fun?

In Div 1 college football there are 119 teams. In the last 2 years 3 former asst coaches from the same University who became head coaches at other schools were fired due to misconduct/mistreatment of players. What are the chances, based on random chance alone, that all 3 coaches were hired from the same school?

Obviously, many head coaches were asst at multiple schools before becoming head coaches. But, usually (as is the case here) they spent the majority of time at one school and are known primarily for that school they were hired from. Let's assume for the sake of this problem that there is only one school they are associated with as an asst coach. I'm trying to make this as simple as possible.

Unfortunately, I'm not exactly sure how many TOTAL coaches were fired for misconduct with players during that time span. So, I'd like to get the probability for four different possibilities (it is almost certainly one of these four), 3 total, 4 total, 5 total, and 6 total. Obviously, if I had the formula I could just insert the four different numbers to get the answer.

Any idea what the formula would be?

The total head coaches is 119 and there are only really 2 positions that a head coach would be taken from, offensive coordinator and defensive coordinator. Of course, when one leaves another takes his place. So over the course of several years there can be several coordinators leaving to become head coaches.

In case it matters, there are 5 head coaches out there culled from this school. However, that's a higher than average amount.

2. Dec 29, 2009

### CRGreathouse

First, a caution about selection bias. You've chosen this team because it has had an unusual number of incidents, but out of 119 teams you'd expect one to have an unusual number of incidents; perhaps this was just the one. Further, you chose this span of years because it was similarly unusual -- not an entirely fair comparison. Similarly, you chose to look at where coaches were assistants because you noticed that as a striking feature, but there may be a number of other features you could have chosen to associate them with schools (their current school, their alma mater, etc.).

Since you have (presumably) selected this team because of the large number of incidents, it's not fair* to ask about that school. (It would be fair to look at that school for the /next/ 2, or even 10 years -- but since I don't have a time machine handy, I'll do it this way instead.) So what we can do is ask if it would be unusual for there to be a team out of the 119 with this many violations. If so, it would be reasonable to conclude that the school is unusual in that respect, apart from the latter two biases I mentioned above. (I'm really not sure how significant that is, since I don't follow football.)

To find a reasonable answer, you'd need to know the number of schools with 3, 4, ... former assistant coaches who were head coach somewhere, over the last 2 years. I have no idea what this might be; an order-of-magnitude estimate is 40 teams with 3, 20 teams with 4, 10 teams with 5, and 5 teams with 6. You'd also need to know the total number of coaches over that period. Clearly this is at least 119 + 3, but it may be much higher depending on turnover.

* That is, it would introduce significant statistical bias.

3. Dec 30, 2009

### hoodleehoo

"So what we can do is ask if it would be unusual for there to be a team out of the 119 with this many violations."

Yes, this is what we are trying to do. :) We are trying to show that this is very unusual and is either an incredible coincidence or that there might be a reason behind the trend. What we want to show is how unusual it is, what the chances of this happening randomly are.

You are really smart! I'll try and see if I can find the information you requested. It won't be easy, but I'll ask around! Thanks!!! :)

4. Dec 30, 2009

### hoodleehoo

Okay, I got the stats. Took a while to dig through every single Div 1 school for the last two years but I got it!

133 total head coaches during the last two years.
2 schools had 5 previous asst's as head coaches during that time.
The remaining seven coaches were never asst's at any school.

Oh, and I can confirm that the 3 mistreatment firings from this one school are the ONLY 3 that occurred during that time period. :)

5. Dec 30, 2009

### CRGreathouse

Great, now I can do an analysis. I can see just by looking at it that this will seem highly relevant -- certainly relevant at the 99% level, compared to the usual 95%. But I caution you about the biases I mentioned above (other than the first): these may simply be unusual years.

So let's pick one of the schools with 5. The chance that 3 were fired from there is 5/133 * 4/132 * 3/131 = 5/191653. There are two such schools, for a total of 10/191653.

Then pick one of the schools with 4. The chance that 3 were fired from there is 4/133 * 3/132 * 2/131 = 2/191653. There are five such schools, for a total of 10/191653. (How coincidental!)

Finally pick one of the schools with 3. The chance that 3 were fired from there is 3/133 * 2/132 * 1/131 = 1/383306. There are twelve such schools, for a total of 6/191653.

All of these events are disjoint -- no two can happen at the same time. This simplifies matters considerably: we can just add probabilities. The total is thus 26/191653, which is about 0.0135% or 1 in 7371.

6. Dec 30, 2009

### hoodleehoo

You are amazing!!! That is so perfect, thanks!

7. Dec 30, 2009

### hoodleehoo

I'm SO SORRY, but I have one more question! Apparently, they did not like how I only used the last school the head coaches attended. So, I need to find the probability based on every school each coach went to. The updated stats are:

ten schools x1
nine x1
eight x2
seven x2
six x4
five x9
four x16
three x21
two x27
one x27

I really hate to ask, but is there anyway you could update it with these figures? I know that's asking a lot and if you can't I understand. But it would really be a life-saver if you were able to!