Winning Baseball Lineup
Winning Baseball Lineup
Fiddler for 8/18/2023, by Laurent Lessard
Expected runs for a full game
Expected runs for a full game
This function computes the average runs for a baseball game with “sluggers” and “hitters”. is the expected runs we will score for the rest of the game if we are currently in inning i,we have o outs, r runners on base, and batter b is up to bat. Function inputs:SLUGGERS: set of sluggers, e.g. {2,4} means sluggers are 2nd and 4th in the batting order.INNINGS: number of innings in the baseball gamepH: hitting probability (probability a hitter will hit a single)pS: slugging probability (probability a slugger will hit a home run)NUMERICAL: (True/False) should Mathematica evaluate the solution numerically (faster), or analytically (slower, but exact)?
V
o,r,b,i
In[]:=
AvgRuns[SLUGGERS_,INNINGS_,pH_,pS_,NUMERICAL_]:=Module[{BATTERS,HITTERS,RUNNERS,OUTS,VARS,E1,E2,E3,E4,E5,EQNS,SOLN,V},(*batters,numberofbases,andnumberofouts*)BATTERS=Range[9];RUNNERS=3;OUTS=3;HITTERS=Complement[BATTERS,SLUGGERS];(*Listofvariables*)VARS=Table[,{o,0,OUTS},{r,0,RUNNERS},{b,BATTERS},{i,INNINGS}]//Flatten;(*iftherearethreeoutsinthe9thinning,gameisover*)E1=Table[==0,{r,0,RUNNERS},{b,BATTERS}];(*ifahitterisupandbasesarenotloaded,weaddarunneroranout*)E2=Table[==pH+(1-pH),{o,0,OUTS-1},{r,0,RUNNERS-1},{b,HITTERS},{i,INNINGS}];(*ifahitterisupwithbasesloaded,weaddarunoranout*)E3=Table[==pH(1+)+(1-pH),{o,0,OUTS-1},{b,HITTERS},{i,INNINGS}];(*ifsluggerhitshomerun,clearthebasesandincrementscore*)E4=Table[==pS(1+r+)+(1-pS),{o,0,OUTS-1},{r,0,RUNNERS},{b,SLUGGERS},{i,INNINGS}];(*3outsinoneinningisthesameas0outsinthenextinningwithclearedbases*)E5=Table[==,{r,0,RUNNERS},{b,BATTERS},{i,INNINGS-1}];(*Listofallequations*)EQNS={E1,E2,E3,E4,E5}//Flatten;SOLN=If[NUMERICAL==True,NSolve[EQNS,VARS],Solve[EQNS,VARS]];(*returnexpectedscorestartingfromfirstbatterinfirstinningwithnooutsorrunnersonbase*)/.SOLN[[1]]];
V
o,r,b,i
V
3,r,b,INNINGS
V
o,r,b,i
V
o,r+1,Mod[b,9]+1,i
V
o+1,r,Mod[b,9]+1,i
V
o,RUNNERS,b,i
V
o,RUNNERS,Mod[b,9]+1,i
V
o+1,RUNNERS,Mod[b,9]+1,i
V
o,r,b,i
V
o,0,Mod[b,9]+1,i
V
o+1,r,Mod[b,9]+1,i
V
OUTS,r,b,i
V
0,0,b,i+1
V
0,0,1,1
Solving special cases
Solving special cases
One inning, one slugger
One inning, one slugger
In[]:=
v=TableAvgRuns{s},1,,,False,{s,9}
1
3
1
10
Out[]=
,,,,,,,,
780011296399952
4236249339922935
892714437481427
4236249339922935
1014032840761142
4236249339922935
3051922249868633
12708748019768805
7729183945570084
38126244059306415
3573996830896157
22875746435583849
3850810161699482
22875746435583849
3986007718407239
22875746435583849
149930373115684
847249867984587
In[]:=
v//N
Out[]=
{0.184128,0.210732,0.23937,0.240143,0.202726,0.156235,0.168336,0.174246,0.176961}
(*makeaprettyplot*)Labeled[BarChart[v,GridLines->{None,Range[0,1,0.05]},ChartLabels->Range[9]],Pane["Average runs scored in one inning as a function of the slugger's position in the batting order",350,Alignment->{Center,Center}],Top,LabelStyle->Directive["Text",TextAlignment->Center]]
Out[]=
Nine innings, two sluggers
Nine innings, two sluggers
In[]:=
(*evaluatenumericallybecausetheexactanalyticsolutionislargeandslowtocompute!*)w=TableAvgRuns{s1,s2},9,,,True,{s1,9},{s2,s1-1};
1
3
1
10
In[]:=
w//TableForm
Out[]//TableForm=
1.93575 | |||||||
1.95816 | 1.98757 | ||||||
1.94247 | 1.97469 | 1.99771 | |||||
1.94107 | 1.94284 | 1.96859 | 1.96591 | ||||
1.92345 | 1.94693 | 1.94794 | 1.94888 | 1.91918 | |||
1.91631 | 1.96187 | 1.98425 | 1.96042 | 1.93426 | 1.90353 | ||
1.91636 | 1.93733 | 1.97991 | 1.9806 | 1.92883 | 1.90123 | 1.92556 | |
1.89841 | 1.92239 | 1.94231 | 1.96654 | 1.93857 | 1.88647 | 1.91387 | 1.9179 |
In[]:=
(*makeaprettyplot*)DiscretePlot3D[w[[i,j]],{i,9},{j,9},PlotRange->{All,All,{1.85,2.00}},Ticks->{Range[9],Range[8],Automatic},AxesLabel->{"Second slugger","First Slugger"},ColorFunction->"SolarColors",ImageSize->Large,ExtentSize->Full,PlotStyle->Opacity[1],PlotLabel->"Average runs scored as a function of sluggers' position in the lineup"]
Out[]=
