There is no general formula - it depends on the performance and results of other teams. You would have to add a lot of assumptions to compute anything, and even in this case the formulas might get quite messy (depending on the assumptions).
I was having fun figuring out the promotion and relegation of my daughter soccer league and ran in to this math question, that i don't even begin to know how to answer.
There a east and a west league. Each team has to play teams in thier league twice and the other league once. There is 16 games per team in the schedule.
Each league has 6 teams, 4 stay up and 2 go down. Points are 1 for a tie and three for a tie, 0 for a loss.
I want to know, what is the formula(and where the variable go) to figure out the number of points the 4th place team must get to stay up. I want the formaula, because the number will change as data changes over the weeks.
For my team, we have played 5 and won 3, tied 1 and lost 1.
I figure, it's a round 22, but want to understand the math. 48 points if we win all 16, we are at 11 know after 5 games.
Thanks for the help.
There is no general formula - it depends on the performance and results of other teams. You would have to add a lot of assumptions to compute anything, and even in this case the formulas might get quite messy (depending on the assumptions).
Typo, but it's clear you mean three for a win.
a) You want to know the lowest point-score that makes you safe. Which is one more than the highest still-possible 5th-place point-score.
b) Also, maybe, you would like to know if you are still in with the slightest chance, because it's yet possible for you (or rather, your daughter's team!) to score more than the lowest still-possible 5th-place point-score.
a) At any time, the highest possible 5th-place point-score will be the highest possible common score of all five teams who are joint-first (or, joint-fifth, depending how you see it). So it is one fifth of the greatest combined score yet achievable by any five teams.
At the start of the contest, this number is 36. (Where each of the five teams wins 6 games within the league and all 6 against the other league too.) So 37 guarantees you stay up.
Further into the game this safety zone might expand downwards, i.e. the danger zone contract, i.e. the highest dangerous score reduce from 36. To find the threshold, find the greatest combined score yet achievable by any five teams, and divide by 5. This may not be too messy if all teams have played the same number of games as each other, in and out of the league. Rank their scores and select the top 5. Add the maximum possible increase to each. Find the mean, i.e. total up and divide by 5. That should be the highest dangerous score.
So you need more information for the update than just your own record. On the other hand, you have a rule of thumb that you want to reach 36 in 16 games, so averaging 2.25 points per game is probably reassuring. You are slightly below that, but until you've updated you don't know that the threshold hasn't come down. (But you do know it hasn't gone up.) Also 'dangerous' only means 'less than 100% safe'.
b) The danger zone also has a lower bound, below which you have no chance at all. At the start, this number is 3. Later on, it's simply the 5th-place score.
This is a lovely puzzle. Of relevance: Tournament (graph theory) - Wikipedia, the free encyclopedia