1. ## Mathematical proof required!

Hi all,

I would like if someone could provide a very brief independent mathematical proof of something. I know it to be true, but others won't accept it, even with my proof.

Imagine a sports league. League A has 12 teams and League B has 16 teams. Each year, the top four teams from each league get promoted to a higher league, League X (so that's eight teams in total).

Now let's imagine that the two leagues merge. We now have a League C, which has two levels, I and II. You get promoted from League C II to League C I, and from League C I to League X. I've tried to describe this pictorially at the end.

Four teams from League C I get promoted to League X, and none from League C II get promoted to League X (they go up to League C I instead).

Problem - show that there is less chance of getting promoted under this new format (probability seems the easiest way of doing this).

Code:
Old format
League X
|
|
--------------------------
|                        |
4 teams                  4 teams
|                        |
League A League B
New format
League X
l
4 teams
l
League C I
l
4 teams
l
League C II

Ultimately, the proof required is that 4 does not equal 8; I'm aware of that and apologise for wasting your collective times here! However, there is someone in cyberland who is making the following argument (and I paraphrase) -

You're just not getting this.

I'm not talking about the total number of promotions across the two leagues, for one very simple reason - no club can qualify for more than one space.

As the league currently stands, Team A can strive for only one of the League A's 4 promotion slots. The fact there are also currently 4 promotion slots in the League B is irrelevant, as Team A can only strive for one of League A's four.

Under the new format - Guess what ? Team A could still only strive for one promotion slot from the combined league's four. Competiton for those slots may possibly be stiffer.

In summary :

Under the old format: 4 promotion slots available to each individual team
Under the new format: 4 promotion slots available to each individual team.

Iin an abstract '4+4=8' fashion, slots have been lost. But in terms of the actual impact on each individual club they haven't been. You can't lose out on something you never had/weren't entitled to in the first place, and no team in League A has ever been entitled to more than one promotion slot from a sum total of four.
(Side note - I suggested to him being in a group of 12 people, four of whom were to be shot. A second group of 16 people came along, of whom also four were to be shot. If they merged, four of the merged group would be shot. Would you merge? QED I thought. Nope. We're dealing with some serious ignorance here. Any help appreciated!)

2. Under the old plan if you are in League A you had 4 in 12 chance of being promoted to League X (33.%), and if you were in League B you had a 4 in 16 chance (25%). You don't say how the teams were alocated to league A or B, but if we assume the teams are randomly assigned that means the overall chance for any one team is 8 in 28 = 28.6%

Under the new structure, your C league has a total of 28 teams, so if only 4 can get promoted to league X. that's a 4 in 28 chance, or 14.3%. So clearly the chance of any one club getting promoted to League X is diminished.

However, you don't address whether moving from C2 to C1 is considered a "promotion." If it is, you could structure your new leagues so that there are 12 teams in C1 and 16 in C2. Now for teams in C1 they have a 4 in 12 chance of getting promoted to X (33.3%), and for teams in C2 they have a 4 in 16 chance of beinfg promoted to C1 (25%). Again, the overal chance of any one team being promoted (to either X or C1) is 28.6%.

Bottom line is you need to decide whether a move from C2 to C1 is a promotion.

3. Nope, C2 to C1 is not a promotion. (Very technically, the question revolved around European qualification for clubs in a new merged league, but I said I'd simplify it for those not acquainted with foreign games!)

I've proven it myself too using sums of probability (note - thread gets very sarcastic and abrasive!!), but it's nice to have independent verification that I'm not going mad.

Thanks!

PS