Originally posted by Lucky Louise
A baseball season is 162 games.
Let 'E(N)' be the number of games you expect the team to win this season.
Let 'S' be the season win total offered by the sportsbook.
Then with the assumption above, the probability of the team's winning any given game (call it 'p') is:
p = E(N)/162.
Using Excel the probability of the team winning more than S games (call it 'Pm') is:
=1-BINOMDIST(S, 162, p, 1)
If S is not an integer, then the probability of the team winning fewer than than S games (call it 'Pf') is:
=BINOMDIST(S, 162, p, 1)
If S is an integer then Pf, the probability of the team winning fewer than than S games, is:
=BINOMDIST(S-1, 162, p, 1)
So the general Excel form for Pf is:
=BINOMDIST(IF(S=INT(S),S-1,S),162,p,1)
Hnece, the fair pricing for Pm>0.5, call it 'F', is:
=-100*Pm/Pf
And the fair pricing F, for Pm<=0.5 is:
=100*Pf/Pm
Which would mean that if the price offered you by the sportsbook is better than F, then you have a good bet.
To compare the two bets, you would just look at the expected value of each and take the higher one iff it's positive.
If 'N' is a season total, and the line is 'L'. Then the expected value of the bet for L>=100 is:
=Pm/(Pf+Pm) * L/100 - Pf/(Pf+Pm)
And the expected value for L<-100 is:
=Pm/(Pf+Pm) * 100/L - Pf/(Pf+Pm)
So to use your example, with over 87.5 -115 versus over 88 +102, and let's say that your handicapping has determined that you expect the team to win 89 games this season.
This means that p = 89/162 = 54.94%
Pm for 87.5/-115 is:
=1-BINOMDIST(87.5, 162, 54.94%, 1)
=59.47%
Pf is:
=BINOMDIST(87.5, 162, 54.94%, 1)
=40.53%
Which means that the expectation with a line of -115 is:
=-Pm/(Pf+Pm) * 100/L - Pf/(Pf+Pm)
=-59.47%/(40.53%+59.47%) * 100/-115 - 40.53%/(40.53%+59.47%)
=11.18%
Pm for 88/202 is:
=1-BINOMDIST(88, 162, 54.94%, 1)
=53.27%
Pf is:
=BINOMDIST(88-1, 162, 54.94%, 1)
=40.53%
Which means that the expectation with a line of +102 is:
=Pm/(Pf+Pm) * 100/L - Pf/(Pf+Pm)
=53.27%/(53.27%+40.53%) * 100/102 - 40.53%/(53.27%+40.53%)
=12.47%
Since 12.47% > 11.18% then, you'd prefer the over 88/+102 to the over 87.5/-115