How Probable Was the BoSox Comeback?

That seems to be the question on everyone's mind today. To have some fun with that question I constructed a simulator in Visual Basic .NET that plays seven games series in a 2-3-2 format. I calculated the probability of each team winning their home games using the log5 method described by Bill James in the 1981 Baseball Abstract and documented by Tom Tippett at Diamond Mind Baseball here.

           A - (A * B) 

WPct = -----------------        

       A + B - (2 * A * B)

So if TeamA played .550 ball at home and TeamB played .450 on the road the probability of TeamA winning a game at home would be .599 (.55-(.55*.45))/(.55+.45-(2*.55*.45)). Interestingly, I also found a slightly different formula by Rodney Sparapani posted this April that gives the same results.

For the 2004 season the Yankees played .704 ball at home, .543 on the road while the Red Sox played .679 at home and .531 on the road. Using the log5 formula for home games then the Yankees had a probablity of winning of .677 against the Red Sox while the Red Sox had a probability of .643 at Fenway Park against the Yankees.

I then used the random number generator to simulate the outcome of the contest and record the number of victories for each side and in how many of the them the team who did not have home field advantage (the Red Sox in this case) one the last 4 games of the series.

Drum roll please....

In 100,000 ALCS matchups the results were:

Yankees: 60,109
Red Sox: 39,891

So the Yankees win about 60% of the time and the Red Sox 40%. In those 100,000 contests the Red Sox won the series after going down 3-0 754 times or .754% of the time or once every 132 series. The Yankees took the first three games 16,701 times and so the Red Sox made their comeback 4.5% of the time. That's very close to the actual number of times that the feat has now been accomplished - 1 out of 26 or 3.8%.

As an aside the Yankees swept the series 5,789 times while the Red Sox swept it 4,269 times. Here is the complete breakdown:

In case you're interested here are the probabilities for the Red Sox facing either the Astros or Cardinals.

Red Sox versus Astros = Red Sox win 60.5% of the time
Red Sox versus Cardinals = Cardinals win 56.3% of the time

For those interested the VB .NET code to run each trial is as follows:

Module Module1



    Public outcomes As New ListDictionary





    Public Sub main()

        outcomes.Add("40", 0)

        outcomes.Add("41", 0)

        outcomes.Add("42", 0)

        outcomes.Add("43", 0)

        outcomes.Add("04", 0)

        outcomes.Add("14", 0)

        outcomes.Add("24", 0)

        outcomes.Add("34", 0)

        Application.Run(New Form1)

    End Sub



    Public Function RunTrial(ByVal team1 As Team, _

      ByVal team2 As Team) As Results



        Randomize()



        Dim team1Won, team2Won As Integer

        Dim team1Home As Decimal

        Dim team2Home As Decimal

        Dim scores As String

        ' odds of team1 winning their home games

        team1Home = (team1.HomeWPct - (team2.AwayWPct * team1.HomeWPct)) / _

         ((team1.HomeWPct + team2.AwayWPct - (2 * team1.HomeWPct * team2.AwayWPct)))

        'team1Home = (team1.HomeWPct * (1 - team2.AwayWPct)) / _

        ' ((team1.HomeWPct * (1 - team2.AwayWPct)) + ((1 - team1.HomeWPct) * team2.AwayWPct))

        ' odds of team2 winning their home games

        team2Home = (team2.HomeWPct - (team1.AwayWPct * team2.HomeWPct)) / _

         ((team2.HomeWPct + team1.AwayWPct - (2 * team2.HomeWPct * team1.AwayWPct)))



        Dim i As Integer

        For i = 1 To 7  '7 game series, 2,3,2



            ' team1 has the homefield advantage



            Select Case i

                Case 1, 2, 6, 7

                    If PlayGame(team1Home) Then

                        ' team1 won

                        team1Won += 1

                        scores &= "1"

                    Else

                        team2Won += 1

                        scores &= "2"

                    End If

                Case 3, 4, 5

                    If PlayGame(team2Home) Then

                        ' team2 won

                        team2Won += 1

                        scores &= "2"

                    Else

                        team1Won += 1

                        scores &= "1"

                    End If

            End Select



            If team1Won = 4 Or team2Won = 4 Then

                ' series over

                Return New Results(team1Won, team2Won, scores)

            End If



        Next

    End Function



    Public Function PlayGame(ByVal prob As Decimal) As Boolean

        ' choose a random number between 0 and 1

        ' if <= prob then game is won so return true else return false

        If Rnd(1) <= prob Then

            Return True

        Else

            Return False

        End If



    End Function



End Module



Public Class Team

    Public Sub New(ByVal Home As Decimal, ByVal Away As Decimal, _

     ByVal name As String)

        HomeWPct = Home

        AwayWPct = Away

        TeamName = name

    End Sub

    Public HomeWPct As Decimal

    Public AwayWPct As Decimal

    Public TeamName As String

End Class



Public Class Results

    Public Sub New(ByVal team1 As Integer, ByVal team2 As Integer, _

     ByVal scores As String)

        Team1Won = team1

        Team2Won = team2

        Games = scores

        outcomes(team1.ToString + team2.ToString) += 1

    End Sub

    Public Team1Won As Integer

    Public Team2Won As Integer

    Public Games As String

End Class