Ah, autumn. The busiest GMAT season of the year as application deadlines and back-to-school nostalgia fill the air, and that season always coincides with Major League Baseball’s pennant races and playoffs. And whether you’re a baseball fan or not, as an aspiring MBA you’ll find a fair amount of overlap between the two, as both the GMAT (and business) and baseball prominently feature the art of probability.
Through that lens, let’s discuss one of the most helpful “tricks” to avoid some of the most time-consuming types of problems on the GMAT, and we’ll lead with a problem:
Whenever his favorite baseball team’s “closer” allows a hit, Sean becomes irate (just close out the game, Joe Nathan!). If the closer needs to get three outs to win the game, and each batter he will face has a .250 batting average (a 1/4 chance of getting a hit), what is the probability that he will give up at least one hit (assuming that there are no walks/errors/hit-batsmen)?
And for those not consumed with baseball, this question essentially asks “if outcome A has a 25% chance of occurring in any one event, what is the probability that outcome A will happen at least once during three consecutive events?”
Baseball makes for an excellent demonstration here, because if we take out the other “free base” situations, really only two things happen – a Hit or an Out. And since we need 3 Outs, we could have all kinds of sequences in which there is at least one hit:
Hit, Out, Out, Out
Out, Hit, Out, Out
Out, Out, Hit, Out
or episodes with multiple hits:
Out, Hit, Hit, Out, Out
Hit, Out, Hit, Out, Hit, Out
Hit, Hit, Hit, Hit, Hit, Hit, Hit, Hit…(game called by mercy rule, Sean punches through his TV)
The GMAT-relevant point is this: when a problem asks you for the probability of “at least one” of a certain event occurring, there are usually several ways that at least one could occur. But look at it this way: the ONLY way that you don’t get “at least one” H is if all three Os come first. The opposite of “at least one” is “none.” And there’s only one way to get “none” – it’s “Not Event A” then “Not Event A” then “Not Event A”… as many times as it takes to finish out the number of events. In other words, in this baseball analogy, if there’s a 25% chance of a hit then there’s a 75% chance of “not a hit” or “Out”, allowing us to set up the ONLY sequence in which there isn’t at least one hit:
Out, Out, Out
Which has a probability of:
3/4 * 3/4 * 3/4
Do the math, and you’ll find that there’s a 27/64 probability of “not at least one hit” and you can then know that the other 37/64 outcomes are “at least one hit.”
To the baseball fan, that means “take it easy on your closer – .250 is a pretty lackluster batting average and that even takes out the chance of walks and errors, and even with *that* there’s a better-than-likely chance there will be baserunners in the 9th!”
To the GMAT student, this example means that when you see a probability question that asks for the probability of “at least one” you should almost always try to calculate it by taking the probability of “none” (which is just one sequence and not several) and subtract that from 1. So your process is:
1) Recognize that the problem is asking for the probability of “at least one” of event A.
2) Find the probability for “not A” in any one event
3) Calculate the probability of getting “not A” in all outcomes by multiplying the “not A” probability as many times as there are outcomes
4) Subtract that total from 1
(and #5 – make sure the problem doesn’t involve any unique probability-changing events like “if outcome A doesn’t happen in the first try then the probability increases to X% for the second try” – that kind of language is rare but does complicate things)
Probability factors into many autumn situations, so whether you’re a GMAT student or a baseball fan, if you know at least this one probability concept your autumn should be a lot less stressful!
By Brian Galvin