By David Freed

Everyone saw it—with the Kansas City Royals down to their last out of October, facing the seemingly un-hittable Madison Bumgarner on the mound, Alex Gordon had a strike of luck: his blooper into center field got past a hard-charging Gregor Blanco, who was unable to put on the breaks as the ball went by him to the wall.

Several seconds later, Gordon was turning hard around second when left fielder Juan Perez fumbled the ball, allowing Gordon to run into third standing up as the ball got sent back to the infield. With a runner on third, Bumgarner held steady, getting Salvador Perez to pop out to end the game and send the Giants to their third World Series in five years.

The question remains: Should Gordon have tried to go home?

**If He Stays**

We can view this situation as an expected value problem. Gordon should have tried to go home if he thought that it would increase his team’s chances of winning. In his analysis, Nate Silver notes that if Gordon gets thrown out at home, it forecloses the possibility of Perez scoring as the winning run with a walk-off homer.

Silver uses a couple shortcuts to get a rough number for this probability, but by mapping out all the future outcomes we can more accurately get a picture of this. Likewise, the analysis by Fangraphs uses its win expectancy metric, which models win expectancy based on the results of previous teams in the situation. For an analysis of this form, it makes more sense to use a more granular model. The Royals, not a sweet hitting team, had the bottom of their order coming up. That’s a relevant factor here and not one we can let slide in our analysis.

From the point at which Gordon decides to go over not, we can map out a set of possible future sets. Using mathematical notation, let’s call them where SK_{i,j,k} represents the Giants’ score, K represents the Royals’ score, and i,j,k are binary indices for whether there is a runner on first, second, or third. Thus we have the following set of outcomes:

*O* = {23_{0,0,1}, 23_{1,0,1}, 23_{1,1,1}, 33_{1,0,0}, 33_{0,1,0}, 33_{0,0,1}, 33_{1,1,0}, 33_{0,1,1}, 33_{1,0,1}, 33_{1,1,1}, K, S, EOI}

where K is the state the Royals win, S is the state the Giants win, and EOI is the state the inning ends in a tie (and sends the game to extras). To calculate the probability of a run, I used the set *O* to create a matrix *P* consisting of the probabilities of going from any one state to another. The element *P*_{12}, for example, consists of the probability you would go from state 1 to state 2—the probability that Bumgarner walks Perez.

To map out the full matrix, we need a matrix (call it *B*) assuming each batter can do six things—single, double, triple, homer, walk, or record an inning-ending out. For the next four batters, Perez, Moustakas, Infante, and Escobar, I found each player’s season average in each category to make *B*. It’s dubious to assume they would have had the same averages against Madison Bumgarner and the San Francisco defense, at this point Bumgarner had thrown nearly 21 innings in three days, making him (theoretically) more hittable and the assumption more tenable. Before continuing, it’s important to note that I assume here that Bumgarner doesn’t come out and Yost doesn’t use a pinch hitter. I also throw out edge scenarios—someone stealing home, a passed ball yielding the win, etc.

1B | 2B | 3B | HR | BB | Out | |

Perez | 0.170 | 0.046 | 0.003 | 0.028 | 0.036 | 0.716 |

Moustakas | 0.12 | 0.042 | 0.002 | 0.03 | 0.07 | 0.736 |

Infante | 0.179 | 0.0365 | 0.005 | 0.010 | 0.057 | 0.711 |

Escobar | 0.194 | 0.0548 | 0.008 | 0.010 | 0.037 | 0.697 |

This matrix maps quickly onto *P*. Once we get *P*, we can backtrack through all the probabilities using the Law of Total Probability. I assume all events are independent—i.e. if Moustakas comes up, he is equally likely to hit a single if Perez is on first or second. Likewise, I assume runners do not advance what is standard (no scoring from second, no going first to third, etc.). As a bottom line, we get the probability that the Royals win in the bottom of the ninth if Alex Gordon stays as 4.76 percent. The probability the Giants win in the bottom of the ninth is 74.5 percent and the probability of a tie is 20.7 percent. Overall, Gordon scores from third about a fourth of the time.

**If He Goes**

The other side of this is what if Gordon took off. Let’s label this decision as G for notation, with G = 1 meaning he was successful and G=0 meaning he was not. If Gordon scores then we have a new state, 33_{0,0,0}. Using our above matrix *P*, we can actually fairly easily calculate the possibilities that the Royals score again in the bottom of the inning. By multiplying this probability by the probability that Gordon successfully gets home, we have a simple conditional probability equation:

P(K|G) = P(G = 1) * P(K|G = 1)

P(K|G) = 2.01%

The probability the inning ends in a tie, if Gordon is successful, is 97.99%.

**Would He Have Made It**

Assuming that extra innings are roughly a coin flip, we can borrow Silver’s math with our more precise numbers and see that the probability of the Royals winning in the case that Gordon stays is 4.76+20.7(.5) = 15.11. This is roughly equivalent to the in-game win probability for the Royals and Silver’s estimate. What if Gordon runs? Then the Royals have a probability of winning of s*(2.01+0.5(97.99)), where s is the probability that Gordon scores. The other numbers represent the probability that the Royals win in the bottom of the ninth—with Perez, Moustakas, and Infante coming up, that’s the above 2.01 percent number. And, of course, if the game goes on we once again assume a 50 percent win probability in extras.

To solve for the optimal probability for Gordon, we set these equal and solve for s , the probability above which he should have run. We get s is approximately 29.6 percent. Silver’s back-of-the-envelope math is thus approximately correct—Gordon needed to be successful in about 30 percent of scenarios for the gamble to be worthy. Silver extends this to say that Gordon should have run, even if the chance was roughly 2-in-1.

Is this realistic? In a word, no.

Screenshots of the game have Crawford getting the ball as Gordon is about to run third and third-base coach Mike Jirschele holds up the stay sign. For the sake of argument, let’s say that we see a windmilling Jirschele waving Gordon around. Gordon has to run basically 100 feet before the Crawford throw hits Posey at the plate. Let’s assume he was at top speed, which is likely 17-20 miles per hour. Then he’d be moving at 20 * (5280/3600) ≈ 30 feet per second. So he’s reaching home in 3.3 seconds, being generous.

How long is Crawford’s throw going to take to reach home? Semi-irrelevant. More important—how fast does Crawford need to throw to beat Gordon? He was a little bit past second (127 feet from home), so roughly 140 feet away. So the number of seconds N it will take to reach home is a function of the speed of the throw V, in miles per hour.

N = 140/V

By adjusting V into feet per second (multiplying by 5280/3600), we can get the following table to see how ridiculously slow Crawford’s throw would have had to be to beat Gordon.

N (sec) | V (mi/hr) |

3.2 | 29.8 |

3.0 | 31.8 |

2.8 | 34.1 |

2.3 | 41.5 |

So Crawford could have beat Gordon home by a second—plenty of time for Posey to apply the tag, especially since Gordon can’t bowl him over—throwing at *41.5* miles per hour. Let’s humor ourselves. If Crawford reaches 70 miles per hour on his throw, probably the most realistic estimate, he gets it home in 1.36 seconds. Gordon has moved 40.8 feet in this time. He’s a sitting duck.

Even if we factor out another second for Crawford to turn and set his feet, Gordon can’t win. The shortstop just has to reach 73 miles per hour to throw him out with a second to spare. Of course, Crawford could miss, but this same shortstop made 21 errors in 634 chances this year, per Baseball-Reference. That’s a 3.3 percent chance—and about the only one Gordon has.

This is roughly a tenth of what Gordon needed. Silver’s math is right, but his logic is flawed. To say this would have been one of the most climactic finishes in baseball history is a nice thought, but the real story ends with Gordon 20 feet from home and Posey ready and waiting for him, a smile on his face and the ball in his glove.

Honestly, as erroneous as the defense was on that at bat, I would include some probability on Posey or Crawford making an error… Gordon should have ran. Go out in a blaze, not a whimper.