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).
New formatCode:Old format League X | | -------------------------- | | 4 teams 4 teams | | League A League B
League C I
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) -
(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!)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.