Baseball Cheat Sheet

by Alex Nelson, 5 September 2015

I found several papers by Jim Alberts, which had a nifty table of probabilities and expected runs in a half-inning. At a recent game, I recalled very few of them, so I put together a “cheat sheet” of probabilities which may be useful in the future.

The data I am using is from 2011-2014. The data for American league (or National league) teams is from games where both the visiting and home teams are in the same league, from 2011-2014.

Expected Runs for a given batter. If a batter comes to plate, and you know which bases have runners, and how many outs there are, here’s the table of expected runs for the batter:

Outs	None	(1)	(2)	(3)	(1,2)	(1,3)	(2,3)	(1,2,3)
0	0.441	0.750	0.936	0.867	1.171	1.091	1.298	1.356
1	0.225	0.420	0.479	0.456	0.599	0.541	0.751	0.708
2	0.071	0.125	0.125	0.115	0.152	0.156	0.197	0.155

For the National League:

Outs	None	(1)	(2)	(3)	(1,2)	(1,3)	(2,3)	(1,2,3)
0	0.426	0.724	0.905	0.839	1.126	1.055	1.198	1.291
1	0.218	0.409	0.475	0.451	0.577	0.547	0.762	0.674
2	0.069	0.121	0.122	0.116	0.140	0.148	0.203	0.154

For the American League:

Outs	None	(1)	(2)	(3)	(1,2)	(1,3)	(2,3)	(1,2,3)
0	0.456	0.775	0.973	0.911	1.230	1.132	1.401	1.455
1	0.231	0.427	0.490	0.459	0.625	0.549	0.755	0.745
2	0.072	0.127	0.130	0.113	0.157	0.163	0.204	0.159

Expected Runs by End of Inning. If you know which bases have runners, and how many outs there are, then we have the following table of expected runs by the end of the inning (a far more useful estimate):

Outs	None	(1)	(2)	(3)	(1,2)	(1,3)	(2,3)	(1,2,3)
0	0.483	0.794	1.199	1.612	1.474	1.955	2.313	2.520
1	0.387	0.706	1.019	1.407	1.266	1.682	1.959	2.154
2	0.372	0.633	0.946	1.004	1.064	1.259	1.362	1.509

If you know the team at bat is in the national league, you may use the following table instead:

Outs	None	(1)	(2)	(3)	(1,2)	(1,3)	(2,3)	(1,2,3)
0	0.466	0.767	1.171	1.566	1.440	1.885	2.222	2.455
1	0.367	0.681	0.991	1.397	1.235	1.645	1.903	2.097
2	0.350	0.599	0.904	0.978	1.039	1.241	1.315	1.456

Likewise, for the American league:

Outs	None	(1)	(2)	(3)	(1,2)	(1,3)	(2,3)	(1,2,3)
0	0.502	0.821	1.234	1.696	1.526	2.051	2.400	2.642
1	0.407	0.729	1.059	1.427	1.304	1.736	2.042	2.230
2	0.391	0.661	0.988	1.047	1.085	1.292	1.404	1.589

The fact the American league has a higher expected-run table probably lies in their designated hitter rule usage.