Run Scored in First Inning
A popular proposition bet in baseball is whether or not a run will be scored in the first inning. It is pretty obvious that the probability of a run in the first inning would be highly correlated to the projected total. The following table shows the number of games with a run in the first inning, according to the total. The second column, titled RSFI, stands for "Run Scored in the First Inning." This table was based on the 2001 to 2011 seasons. The lower right cell shows an overall probability of at least one run in the first inning of 52.5%.
[TABLE="class: data"]
<tbody>[TR]
[TH="class: title, bgcolor: #000000, colspan: 4, align: center"]Run Scored in First Inning Data[/TH]
[/TR]
[TR]
[TH="bgcolor: #BBBBBB"]Total[/TH]
[TH="bgcolor: #BBBBBB"]RSFI[/TH]
[TH="bgcolor: #BBBBBB"]Total Games[/TH]
[TH="bgcolor: #BBBBBB"]Ratio[/TH]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]0[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1[/TD]
[TD="bgcolor: #FFEEAA, align: right"]0.00%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]5.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]13[/TD]
[TD="bgcolor: #FFEEAA, align: right"]17[/TD]
[TD="bgcolor: #FFEEAA, align: right"]76.47%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]6[/TD]
[TD="bgcolor: #FFEEAA, align: right"]30[/TD]
[TD="bgcolor: #FFEEAA, align: right"]68[/TD]
[TD="bgcolor: #FFEEAA, align: right"]44.12%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]6.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]134[/TD]
[TD="bgcolor: #FFEEAA, align: right"]335[/TD]
[TD="bgcolor: #FFEEAA, align: right"]40.00%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]7[/TD]
[TD="bgcolor: #FFEEAA, align: right"]537[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1127[/TD]
[TD="bgcolor: #FFEEAA, align: right"]47.65%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]7.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1093[/TD]
[TD="bgcolor: #FFEEAA, align: right"]2313[/TD]
[TD="bgcolor: #FFEEAA, align: right"]47.25%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]8[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1544[/TD]
[TD="bgcolor: #FFEEAA, align: right"]3109[/TD]
[TD="bgcolor: #FFEEAA, align: right"]49.66%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]8.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]2603[/TD]
[TD="bgcolor: #FFEEAA, align: right"]5039[/TD]
[TD="bgcolor: #FFEEAA, align: right"]51.66%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]9[/TD]
[TD="bgcolor: #FFEEAA, align: right"]3054[/TD]
[TD="bgcolor: #FFEEAA, align: right"]5859[/TD]
[TD="bgcolor: #FFEEAA, align: right"]52.12%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]9.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]2415[/TD]
[TD="bgcolor: #FFEEAA, align: right"]4403[/TD]
[TD="bgcolor: #FFEEAA, align: right"]54.85%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]10[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1199[/TD]
[TD="bgcolor: #FFEEAA, align: right"]2163[/TD]
[TD="bgcolor: #FFEEAA, align: right"]55.43%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]10.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]811[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1358[/TD]
[TD="bgcolor: #FFEEAA, align: right"]59.72%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]11[/TD]
[TD="bgcolor: #FFEEAA, align: right"]313[/TD]
[TD="bgcolor: #FFEEAA, align: right"]516[/TD]
[TD="bgcolor: #FFEEAA, align: right"]60.66%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]11.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]128[/TD]
[TD="bgcolor: #FFEEAA, align: right"]206[/TD]
[TD="bgcolor: #FFEEAA, align: right"]62.14%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]12[/TD]
[TD="bgcolor: #FFEEAA, align: right"]72[/TD]
[TD="bgcolor: #FFEEAA, align: right"]112[/TD]
[TD="bgcolor: #FFEEAA, align: right"]64.29%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]12.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]64[/TD]
[TD="bgcolor: #FFEEAA, align: right"]103[/TD]
[TD="bgcolor: #FFEEAA, align: right"]62.14%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]13[/TD]
[TD="bgcolor: #FFEEAA, align: right"]47[/TD]
[TD="bgcolor: #FFEEAA, align: right"]68[/TD]
[TD="bgcolor: #FFEEAA, align: right"]69.12%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]13.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]26[/TD]
[TD="bgcolor: #FFEEAA, align: right"]41[/TD]
[TD="bgcolor: #FFEEAA, align: right"]63.41%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]14[/TD]
[TD="bgcolor: #FFEEAA, align: right"]10[/TD]
[TD="bgcolor: #FFEEAA, align: right"]16[/TD]
[TD="bgcolor: #FFEEAA, align: right"]62.50%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]14.5[/TD]
[TD="bgcolor: #FFEEAA, align: right"]8[/TD]
[TD="bgcolor: #FFEEAA, align: right"]12[/TD]
[TD="bgcolor: #FFEEAA, align: right"]66.67%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]15[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1[/TD]
[TD="bgcolor: #FFEEAA, align: right"]1[/TD]
[TD="bgcolor: #FFEEAA, align: right"]100.00%[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]Total[/TD]
[TD="bgcolor: #FFEEAA, align: right"]14102[/TD]
[TD="bgcolor: #FFEEAA, align: right"]26867[/TD]
[TD="bgcolor: #FFEEAA, align: right"]52.49%[/TD]
[/TR]
</tbody>[/TABLE]
Here is what that looks like in a graph. The blue line shows the actual percentage of games with a run in the first inning by estimated total runs, from 6.5 to 11.5. The red line is called a "least-squared" regression line, which smooths out the ups and downs by showing the line that best matches the game total to the probability of a first-inning run. The equation for that line is p=0.2554 + 0.0304×t, where p = probability of first-inning run, and t=estimated total runs.
I also considered whether it mattered if the game had a big favorite. Given the same total, I found that that money lines on the game didn't affect the probability of a first-inning run.
In baseball, the odds are often different depending on whether a bet is on the under or over the projected total runs. For example, in a game on April 13, 2012, between the Brewers and Braves the total is 7.5. At Bovada, the line on the over is -125, and on the under is +105. Given the pressure on a high total, that would indicate an increased probability of a run in the first inning. What I suggest to adjust for that is to use the following modified formula when there is unequal demand between the over and under.
p = 0.2554 + 0.0304×t + 0.000724×f, where
p = Probability of first-inning run
t = Projected game total
f = Favoritism points on the over
Let me explain "favoritism points on the over." This is how many point the money line on the over has moved in the direction of the over being the favorite side. For example, if the sports book has a 20-cent line on the total, meaning the under and over are 20 points apart, then the following table shows the number "favoritism points on the over."
[TABLE="class: data"]
<tbody>[TR]
[TH="class: title, bgcolor: #000000, colspan: 4, align: center"]Favoritism Points
on the Over[/TH]
[/TR]
[TR]
[TH="bgcolor: #BBBBBB"]Money Line
on Over[/TH]
[TH="bgcolor: #BBBBBB"]Favoritism
Points[/TH]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]130[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-40[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]125[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-35[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]120[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-30[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]115[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-25[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]110[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-20[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]105[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-15[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]100[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-10[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-105[/TD]
[TD="bgcolor: #FFEEAA, align: right"]-5[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-110[/TD]
[TD="bgcolor: #FFEEAA, align: right"]0[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-115[/TD]
[TD="bgcolor: #FFEEAA, align: right"]5[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-120[/TD]
[TD="bgcolor: #FFEEAA, align: right"]10[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-125[/TD]
[TD="bgcolor: #FFEEAA, align: right"]15[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-130[/TD]
[TD="bgcolor: #FFEEAA, align: right"]20[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-135[/TD]
[TD="bgcolor: #FFEEAA, align: right"]25[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-140[/TD]
[TD="bgcolor: #FFEEAA, align: right"]30[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-145[/TD]
[TD="bgcolor: #FFEEAA, align: right"]35[/TD]
[/TR]
[TR]
[TD="bgcolor: #FFEEAA, align: right"]-150[/TD]
[TD="bgcolor: #FFEEAA, align: right"]40[/TD]
[/TR]
</tbody>[/TABLE]
Let's look at a couple of examples, based on a 20-cent total.
Example 1: Mets vs. Phillies, over 6.5 -115. The table above shows 5 favoritism points. So the probability of a run in the first inning is:
p = 0.2554 + 0.0304×6.5 + 0.000724×5 = 0.45662. That equates to a fair line on a first-inning run of +119.
Example 2: Orioles vs. Yankees, over 7.5 -105. The table above shows -5 favoritism points. So the probability of a run in the first inning is:
p = 0.2554 + 0.0304×7.5 + 0.000724×-5 = 0.47978. That equates to a fair line on a first-inning run of +108.
If you're using a sports book with other than a 20-cent line on the total, then adjust the favoritism points accordingly.
A look at actual lines on this prop shows that most sports books set them well enough to not find an advantage either way. However, I found one book that seemed to favor the "no" on this prop, often making betting on a first-inning run a good value. It is good to try to bet at such places early, to beat the other sharp bettors.