## Thursday, August 26, 2010

### Baseball

Today the Tigers are 64 and 64, and officially 8.5 games back.  Do they have a chance?  Lets calculate mathematically the chance.

Current standings:
 AL - Central W L GB PCT 72 55 - 0.567 69 58 3 0.543 64 64 8.5 0.500

The baseball season has 162 games, that leaves 162 - 64 - 64 or 34 games left in which to make up their ground with the Twins.  They have 162 - 72 - 55 or 35 more games.  Suppose we are fortunate enough to find the Twins going just under .500 the rest of the season -- 17 wins and 18 losses.  They would finish with a 89 - 73 season.  We'd have to win 89-64 = 25 games out of 34 left, or .735 the rest of the season.  The best teams in baseball are the Rays and Yankees and they have an average of only .614.  Only two teams in baseball history have had .735 seasons -- the 1906 Cups and 1902 Pirates.  Granted, we're not asking the Tigers to play .735 for an entire season, but still....

An interesting mathematical challenge: Calculate the required Tigers percentage for a given Twins percentage x.  I think 35*x is the number of additional Twins wins, and so 72+35x would be their final wins.  (72+35x) - 62 would be our required wins, and so T = (72+35x)-62  /34 should do it I think.
Here's a graph:

The x-axis is the Twins winning percentage.
The y-axis is the Tigers winning percentage.

What does the y-intercept represent on our graph?
What does the point where the graph leaves off the top of the page represent?

What would the point (0.2, 0.8) represent?  Who would be in the playoffs?  Can you find other points with the same results? Where are they all located?  If you could shade all the points that would allow the Tigers into the playoff, where would the shading be?  Approximately what percentage of the possible area would be shaded?  That's the likelihood that the Tigers can make the playoffs.

Every day this graph changes -- and if the status quo is maintained, the line will move ever more up and to the right.  What does that mean for the Tigers?