October 19, 2005

Lies, Damned Lies

Running the Odds

by Nate Silver

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A thousand-to-one shot. That was my first reaction when the Cardinals rallied to beat the Astros on Monday night, down to their last strike with a two run-deficit against one of the scariest one-inning pitcher in baseball.

Actually, that was my second reaction. My first reaction was something that would have made Andy Pettitte blush. And my third reaction, after a few moments of reflection about Donnie Moore, Grady Little, and Steve Bartman, was that we might have just witnessed the most unlikely comeback in postseason history.

We don't have a comprehensive way to test that proposition. There is probably some long-forgotten game from the early days of baseball when the Boston Pilgrims scored 13 times in the bottom of the ninth to shock the Cincinnati Red Stockings. But we can at least compare Monday's events against some other recent postseason comebacks, by estimating the odds against the trailing team coming back to win the game when facing a seemingly dire situation. These estimates will be based on three things:

1. The Expected Wins Matrix. This gives us the empirical probabilities of a team winning the game given a particular inning, score differential, number of outs and number of baserunners. We will use 10 years worth of data to come up with our estimates, since these things are subject to sample-size considerations, particularly for things like the probability of an unlikely victory.

2. The opposing pitcher. The most important thing for a team to do when down to its last few outs is to extend the inning. Thus, we'll look at the opposing pitcher's OBP against for the season in question. We'll assume that an "average" pitcher allows a OBP of .300. This is deliberately lower than the actual league OBP, which usually runs in the .330 range. The Expected Wins statistics are compiled from historical gameplay, and a team down in the late innings will usually be facing the closer, another good relief pitcher, or an ace starter who has stayed in the game because he's throwing effectively. We'll multiply the raw win probability by the ratio of the opposing pitcher's OBP to the "league average" of .300 to come up with an adjusted figure. For example, if a team would ordinarily have a 6% probability of coming back to win the game, but they're facing a tough pitcher with an OBP allowed of .250, the modified figure will be (6% * (.250/.300)), or 5%.

3. The count. Historical ball-strike records are hard to come by, but we'll make a further adjustment for the count in the two games in which it was most important: Dave Henderson's home run against Moore, which came after an 0-2 count, and the Cards' comeback on Monday.

The final step is translating the adjusted win probability into some poker-style odds, which is straightforward. Counting down seven memorable finishes, in increasing order of improbability:

7. Red Sox @ Yankees, ALCS Game 7, 2003.

• The Situation. Yankees trailed by three runs with one out in the bottom of the eighth before Derek Jeter doubled. Note that the goal here is to freeze the moment in time at which the comeback was least likely. We don't look at the probability of the Yankees winning the game when Aaron Boone hit his home run, by which time the Yankees, as the home team against a depleted bullpen, were the favorites. We look at the events that made the climactic finish possible.

• Win Probability. The home team won 4.12% of the time when this situation occurred between 1996-2005.

• Pitcher Adjustment. Pedro Martinez allowed a .271 OBP in 2003. Some observers have commented that Martinez wasn't pitching his best baseball at the time, but since this situation isn't a serious contender for Greatest Comeback, we'll leave it at that. Probability adjusted downward to 3.72%.

• Odds: 26-1 against.

6. A's @ Dodgers, World Series Game 1, 1988.

• The Situation. Down by one in the bottom of the ninth, the Dodgers were down to their last out before Mike Davis walked, eventually setting up Kirk Gibson's home run. This was against Dennis Eckersley in one of his best seasons.

• Win Probability. The home team won 3.52% of the time when this situation occurred between 1984-1993.

• Pitcher Adjustment Eckersley allowed a .230 OBP in 2003. Probability adjusted downward to 2.70%.

• Odds: 36-1 against.

5. Marlins @ Cubs, NLCS Game 6, 2003.

• The Situation. Cubs led by three runs with one out in the top of the eighth before Juan Pierre doubled to left. A consultant named Steve Bartman sat in the third-base boxes with his headphones on, regretting that he had not ordered a beer before last call.

• Win Probability. The visiting team won 2.49% of the time when this situation occurred between 1996-2005.

• Pitcher Adjustment. Mark Prior's OBP against in 2003 was .254. Probability adjusted downward to 2.11%.

• Odds: 46-1 against.

• Bonus. The Bartman Ball had occurred immediately after Pierre's double, which had improved the Marlins' adjusted win probability to 4.04, or 24-to-1 against.

4. Braves @ Astros, NLDS Game 4, 2005

• The Situation. The Astros trailed by five runs to start the eighth inning of the impromptu doubleheader before Brad Ausmus led off with a walk.

• Win Probability. The visiting team won 2.32% of the time when this situation occurred between 1996-2005.

• Pitcher Adjustment. Tim Hudson started the inning, but the vast majority of the damage was done against Kyle Farnsworth, so we'll use his figure instead. Farnsworth's OBP allowed this year was .261. Probability adjusted downward to 2.02%.

• Odds: 49-1 against. A late-inning, five-run comeback is remarkable by any standard. But it's worth noting that the number of outs remaining--the Astros had six left--is generally a more important constraint than the deficit.

3. Red Sox @ Angels, ALCS Game 5, 1986.

• The Situation. The Red Sox trailed by three runs to start the ninth inning. That's as bad as things got. Bill Buckner got the ball rolling immediately by leading the inning off with a single, and while the Red Sox came down to their final strike before Henderson hit the home run that would haunt Donnie Moore for the rest of his life, by that point the gap had been closed to one run and the Red Sox had the tying run on first base.

• Win Probability. Visiting Team won 1.86% of the time when this situation occurred between 1982-1991.

• Pitcher Adjustment. Moore allowed a .282 OBP in 1986. Probability adjusted downward to 1.74%.

• Odds: 56-1 against.

• Bonus. Henderson came to the plate as the go-ahead run with a runner on first and two outs. The home team won 9.01% of the time when that situation occurred between 1982-1991. However, Moore had run the count to 0-2 against him. While we don't have detailed split stats on Moore, roughly speaking, a hitter's OBP is cut in half when this situation occurs. So we'll assume a .141 OBP against him after an 0-2 count, reducing win probability to 4.23%, for odds of 23-1 against.

2. Red Sox @ Mets, World Series Game 6, 1986.

• The Situation. The Mets were trailing by two, down to their last out in their last inning (in this case, the tenth rather than the ninth) before Gary Carter singled. This situation is rather similar to the one that we witnessed on Monday. However, the Mets were the home team, while the Cardinals were the visiting team, so the Red Sox had no last chance to salvage Bill Buckner's dignity. Also, Calvin Schiraldi, while he hails from Houston, cannot be mistaken for Brad Lidge.

• Win Probability. We'll cheat by looking at results from the ninth inning rather than the tenth, since extra-inning games occur too rarely to provide for a reliable sample size. The home team won 1.12% of the time when this situation occurred in the ninth between 1982-1991.

• Pitcher Adjustment. Schiraldi was a tough cookie in 1986, allowing a .265 OBP against. Probability adjusted downward to 0.99%.

• Odds: 100-1 against.

• Bonus. This is remembered as a "last strike" game, but the last strike did not occur against Carter. It occurred against Ray Knight, who was down 0-2 after Carter and Kevin Mitchell had reached base. The Mets started the at bat with a 6.99% adjusted win probability, which was reduced to 3.49% once Knight had two strikes against him, for 28-1 odds against.

1. Cardinals @ Astros, NLCS Game 5, 2005

• The Situation. One ball, two strikes, David Eckstein facing Brad Lidge.

• Win Probability. Even before accounting for Lidge and the two-strike count, this is a nearly impossible spot. No visiting team came back from a two-run deficit when down to their last out in the entire 2003 season. Same thing was true in 2004. The overall win probability in this situation between 1996-2005 was 0.76%.

• Pitcher and Count Adjustment. The OBP against Lidge after a 1-2 count was .132 this season. That figure doesn't appear to be a fluke; he's been in the .150 neighborhood for each of the past four years. Probability adjusted downward to 0.33%.

• Odds: 298-1 against.

Not quite a thousand-to-one, but longer odds, say, than USC mounting a successful two-minute drive against Lidge's alma mater. Clay's Postseason Odds Report estimates that the Cards have a 29.4% chance of going on to win the series from this point onward. If we parlay that against the probability of their coming back to win Game 5, we come up with a figure of 0.098%, or 1017-1 odds against.

Of course, the Cardinals haven't won the series just yet. It's interesting to note, however, that the major sports futures Web sites treat their chances more favorably than Clay does; as of this writing, their odds are on the order of 36%. For a variety of reasons, I don't go anywhere near betting sports, but I don't think I'd take the short side of that bet. The Cardinals are coming off what is probably the most unlikely comeback in postseason history. If there were ever a series governed by momentum, morale, destiny, and the rest of the things that statheads hate to talk about, it is this one.

Nate Silver is an author of Baseball Prospectus. 
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