Recreational Math – Baseball (Top of 9th, Man on 1st, 0 Out)

February 12, 2010 at 8:44 am Leave a comment

Baseball was the game of my youth. I spent countless summer days playing wiffle ball from dusk until dawn, taking a short break for a sandwich, the ice cream man, and the Del’s frozen lemonade truck. (I can still taste the Del’s! in my mind!) Growing up in Rhode Island, I was also a die hard Red Sox fan. When I woke up, I read the sports section recap while I ate and closed my day listening to the game on the radio.

Baseball is also the game of my adulthood, but for different reasons. First, the Red Sox have managed to win a pair of World Series titles. Winning is good. Second, there are so many applications of math, probability, and statistics in baseball. Some credit the use of sabermetrics with helping the Sox to win, both by the front office (building the team) and the manager (making game decisions based on data and probabilities).

No Outs, Top of the Ninth, Man on First – What To Do?

In today’s article I examine an issue of contention between baseball purists and sabermetricians, tying it to probability that can be used in an Intro Stats course. Here’s the situation – the game is tied heading to the top of the 9th inning. The visiting team manages to get its leadoff hitter on first base. Should he try to steal second base? Should the next hitter try a sacrifice bunt to advance the runner to “scoring position” at second base? Let’s analyze the options.

Win Expectancy

I am basing my analysis on “Win Expectancy”. Win Expectancy is the probability of a team winning a game based on that game’s “state” – inning, outs, baserunners. The website I am using ( to calculate these probabilities coded games from 1997 through 2006, counts how many games were in that state, and the percentage of times that the home team went on to win the game.

As the inning starts, the visiting team has a 47.8% chance of winning the game. (If the visiting team fails to score, that chance drops to 34.4%.) If the first batter reaches 1st base safely, the win expectancy rises by nearly 9% to 56.7%.

Should the Runner Try to Steal 2nd Base?

If the runner successfully steals 2nd base, the chances for victory jump to 65.7%. If he is caught stealing, the chances for victory drop to 42.0%. Should he steal? It depends on his chances of success.

Let “p” represent the probability of stealing 2nd base and 1 – p represent the probability of getting thrown out. The expected value for the Win Expectancy can be expressed as follows.

0.657 p + 0.420 (1 – p)

This simplifies to 0.237 p + 0.420. Stealing makes sense if the expected value of Win Expectancy increases from its original state of 56.7%.

0.237 p + 0.420 > 0.567

p > 0.620

Win Expectancy increases if the player has a greater than 62% chance of successfully stealing 2nd base. For what it’s worth, the total success rate in 2008 was 73%, but you have to consider that most often stolen bases are attempted by the top base runners and not your average runner. (A look at run expectancy suggests that the percentage needs to be above 75% to be beneficial in general. Perhaps I’ll address that in the future.)

Should the Next Batter Try to Sacrifice the Runner to 2nd Base?

There are 3 likely outcomes when a batter attempts a sacrifice bunt: runner safe/batter out (61.7% of the time during the 2003 season), runner out/batter safe (23.5%), and runner safe/batter safe (14.8%).

Here are the Win Expectancy values for these situations: runner safe/batter out (55.6%), runner out/batter safe (48.3%), and runner safe/batter safe (66.5%).

Let’s look at the expected value of Win Expectancy if the batter attempts a sacrifice bunt.

0.617 (0.556) + 0.235 (0.483) + 0.148 (0.665) = 0.555

So, the sacrifice bunt lowers the Win Expectancy from 56.7% to 55.5%, telling us that the sacrifice bunt hurts your chances of winning in this scenario.


Of course, the manager’s decision depends on the runner (is he an above average base stealer?) and the man at bat (is he a skilled bunter?). If you estimate the runner’s chances of a successful steal to be above 62%, the move increases your chance for victory. The sacrifice bunt hurts your chances for victory. I guess that’s why Dave Roberts stole 2nd in the legendary 4th game of the 2004 ALCS instead of having Bill Mueller attempt the sacrifice bunt. It was just a simple matter of mathematics!


I am a math instructor at College of the Sequoias in Visalia, CA. You can reach me through the contact page on my website –


Entry filed under: Math, Recreational Math. Tags: , , , , , , , , , , , , , .

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