GMAT Tip of the Week: Baseball Probability is Probably Stressful

The Major League Baseball League Championship Series begin tonight, and for avid fans, stat geeks, and yes aspiring MBA students, there will be probability lessons abound. Baseball is the ultimate probability-and-statistics sport – the book/movie Moneyball was all about probability; Nate Silver, he of presidential election prediction models, first fell in love with statistics as a baseball fan. So whether you’re living and dying with every pitch or you don’t understand why your roommate is, what happens at Fenway Park tomorrow night can help you to increase your probability of going to school just across the Charles at MIT or Harvard.

Suppose, for example, you’re both a Detroit Tigers fan and a GMAT scholar. You entered the 9th inning in last night’s winner-take-all game against the Oakland A’s with a 3-run lead. The last inning is just a victory lap, right?

Wrong. Your author, both a Tigers fan and a GMAT scholar, can attest to the stress associated with closing out those last three outs. Why? Because of GMAT-style probability.

If your team is trying to close out a 9th inning, your stress level rises dramatically with each baserunner. Each runner represents a potential run, particularly if the batter hits a home run. So ideally your team won’t allow any baserunners, because any time a man reaches base you get a knot in your stomach and the next at-bat becomes all the more stressful. But wait – the average hitter in MLB bats something like .250-.270, meaning he gets a hit every 25 to 27% of the time. There’s a pretty low probability that these guys will get hits, so you should rest comfortably, right? Here’s where GMAT probability comes in – no. Even if we take the low end of that average, that 25% of the time each batter will get a hit (and for the sake of simpler math we’ll leave out walks and hit-by-pitch), in order to get three straight outs you need one specific sequence: Out, Out, Out.

In GMAT probability problems, this need for a specific sequence comes up often (for example, “what is the probability that it will be sunny three days in a row” or “what is the probability that Alice and Brittany each pass the test but Charlie does not”). When this happens, you have to multiply together the probabilities of each outcome in that sequence. And for us to get three straight outs, we’ll take the 1/4 probability that the batter gets a hit and recognize that the other 3/4 of the time he does not. So the probability is:

3/4 * 3/4 * 3/4 = 27/64. There’s a less than 50% chance that – even if your pitcher is facing three below-average hitters – you’ll get out of the inning without any baserunners. And say that the second hitter is actually better than that, batting .300. Facing him you only have a 7/10 chance of getting him out, so the new probability is:

3/4 * 7/10 * 3/4 = 63/160, which is about a 38% chance of getting out of the inning without a baserunner. It is very likely that you’ll be stressed!

Now, these are all the probabilities for “no baserunners / everyone out”, an outcome that you’d want if your team is pitching and holding on to a lead. But say you’re in the opposite situation, trailing and rooting for your team to get a hit. Every baserunner gives you that much more hope, so you’re rooting for hits. What is the probability that you get *at least one* baserunner?

This “at least one” probability is a very commonly-tested GMAT structure, and can be tricky. Even with just three hitters, all batting .250, the process of figuring out all of the situations that will give you at least one hit is tricky. It could be:

Hit, No Hit, No Hit
Hit, Hit, No Hit
No Hit, Hit, Hit
No Hit, No Hit, No Hit

And particularly if you’re stressed – either watching a 9th inning battle in the playoffs or taking the GMAT under timed conditions – you’re liable to forget one sequence (what about Hit, Hit, Hit?) or botch a calculation. But there’s a trick – there’s only one way to NOT get “at least one” hit. That’s “no hits”. And since there’s a 100% total probability (the probability that “some outcome” will happen), the probability of “anything but no hits” is 100% minus the probability of “no hits”. So if there’s a 27/64 probability of no hits with your mediocre lineup of all .250 hitters, there’s a 64/64 – 27/64 = 37/64 chance of getting a hit in that inning and keeping your hopes alive. And the lesson – other than Yogi Berra’s famous “it ain’t over til it’s over” – is that if you’re asked for the probability of “At Least 1”, it’s usually much more efficient to find the probability of “none” and subtract from 100%.

Even more broadly, recognize this – if you’re passionately pulling for the Tigers, Red Sox, Cardinals, or Dodgers this next week, you probably won’t have much mental energy to turn it into much GMAT homework. Just know that if your reliever puts a runner on base, he’s not a worthless pile of garbage…the probability was in favor of that; and know that if your team is trailing there’s always a chance. But if your roommate/boyfriend/coworker is passionately watching these games and you’re not quite as invested, recognize that it’s a great opportunity to think about sequential probability and practice for the GMAT. It worked for Michael Lewis and his bestseller Moneyball; it worked for Brad Pitt the same way; it worked for Nate Silver and his prob/stats empire; and it can work for you. To master GMAT probability, play ball.