Magic number (sports)
From Wikipedia, the free encyclopedia
- For other uses of the term, see magic number
In certain sports, a magic number is a number used to indicate how close a front-running team is to clinching a season title. It represents the total of additional wins by the front-running team or additional losses by the rival team after which it is mathematically impossible for the rival team to capture the title in the remaining games. This assumes that each game results in a win or a loss, but not a tie. Teams other than the front-running team have what is called an elimination number (or "tragic number") (often abbreviated E#). This number represents the number of wins by the leading team or losses by the trailing team which will eliminate the trailing team. The elimination number for the second place team is exactly the magic number for the leading team.
The magic number is calculated as G + 1 − WA − LB, where
- G is the total number of games in the season
- WA is the number of wins that team A has in the season
- LB is the number of losses that team B has in the season
For example, in Major League Baseball there are 162 games in a season. Suppose the top of the division standings late in the season are as follows:
| Team | Wins | Losses |
| "A" | 96 | 58 |
| "B" | 93 | 62 |
Then the magic number for team "A" to win the division is 162 + 1 − 96 − 62 = 5.
Any combination of wins by team "A" and losses by team "B" totalling to 5 makes it impossible for team "B" to win the division title.
The "+1" in the formula serves the purpose of eliminating ties; without it, if the magic number were to decrease to zero and stay there, the two teams in question would wind up with identical records. If circumstances dictate that the front-running team would win the tiebreaker regardless of any future results, then the additional constant 1 can be eliminated. For example, the NBA uses complicated formulae for breaking ties, using several other statistics of merit besides overall win/loss record; however the first tiebreaker between two teams is their head-to-head record; if the frontrunning team has already clinched the better head-to-head record, then the +1 is unnecessary.
In other sports, such as major league baseball, ties are usually broken by an additional play-in game(s) between the teams involved; The additional +1 cannot be removed from the formula unless both teams are in the same division and one has clinched the season series and both are already guarnteed to make the playoffs (one as a division champion the other as wildcard). When a team gets to the point where its magic number is 1, it is said to have "clinched a tie" for the division or the wild card. However, if they end the season tied with another team, and only one is eligible for the playoffs, the extra playoff game will erase that "clinching" for the team that loses the playoff game.
[edit] Derivation
The formula for the magic number is derived straightforwardly as follows. As before, at some particular point in the season let team "A" have WA wins and LA losses. Suppose that at some later time, team "A" has wA additional wins and lA additional losses, and define similarly WB, LB, wB, lB for team "B". The total number of wins that team "B" needs to make up is thus given by (WA + wA) − (WB + wB). Team "A" clinches when this number exceeds the number of games team "B" has remaining, since at that point team "B" cannot make up the deficit even if team "A" fails to win any more games. If there are a total of G games in the season, then the number of games remaining for team "B" is given by G − (WB + wB + LB + lB). Thus the condition for team "A" to clinch is that (WA + wA) − (WB + wB) = 1 + G − (WB + wB + LB + lB). Cancelling the common terms, we obtain wA + lB = G + 1 − WA − LB, which establishes the magic number formula.
[edit] Subtlety
Sometimes a team can appear to have a mathematical chance to win even though they have actually been eliminated already, due to scheduling. In this major league baseball scenario, there are three games remaining in the season. Teams "A", "B" and "C" are assumed to be eligible only for the division championship; another team with a better record in another division has already clinched the one available "wild card" spot:
| Team | Wins | Losses |
| "A" | 97 | 62 |
| "B" | 97 | 62 |
| "C" | 95 | 64 |
If Team "C" were to win all three remaining games, it would finish at 98-64, and if both Teams "A" and "B" were to lose their three remaining games, they would finish at 97-65, which would make Team "C" the division winner. However if Teams "A" and "B" are playing against each other in the final weekend (in a 3 or 4 game series), one of them will necessarily have to win at least two games and thereby clinch the division title with a record of either 100-62 or 99-63. The more direct consequence of this situation is that it is also not possible for Teams "A" and "B" to finish in a tie with each other.
