GMAT Tip: A Tricky Probability Question

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This tip on improving your GMAT score was provided by Brian Galvin at Veritas Prep.

For many GMAT test-takers, probability looms as one of the more daunting concepts on the exam. While the majority of the skills tested on the GMAT—particularly arithmetic, algebra, and geometry—are standard features of the typical K-12 math curriculum, probability is a subject that many students haven’t studied directly in school. So many GMAT test-takers lie awake nights trying to predict the likelihood (should we say probability?) that they’ll have to face more than one probability question on test day.

To make it even more daunting, the authors of the GMAT love to use a particular word that makes probability look that much trickier: simultaneous.

You can quickly learn to master basic probability questions such as “What is the probability of getting ‘tails’ twice in a row on two consecutive flips of a coin?” (It’s 1 out of 4, as there are four total outcomes: HH, HT, TH, and TT.) But the word “simultaneous” adds a new, multi-syllable wrinkle to these problems.

Well, rest assured that this tricky word is just that: a trick. There is no such thing as “simultaneous.”

Consider the question:

A jar contains ten marbles, five of which are red and five of which are blue. If Kori reaches in the jar and simultaneously picks two marbles at random, what is the probability that both will be red?

That word “simultaneously” adds an element of specificity to the question that really doesn’t need to be there. Consider:

• Even if Kori pulls two marbles out of the jar at the same time, her hand will inevitably touch one before touching the other.

• Or one of them will leave the jar before the other does, unless she holds them perfectly horizontally.

• Or her eyes will inevitably recognize one of them before they recognize the other as she’s looking to see what colors they are.

• Or even if all the above three events happen, the human brain has to process them individually in some kind of sequence.

Which is all just a long way of saying that “simultaneous” events—at least for the purposes of GMAT probability—do not happen simultaneously. You can always assign an order to them, so to calculate them you should calculate the sequences that get you to your desired outcome. Here, just one sequence is possible: Red, then Red.  You’d calculate the probability of red on the first draw (five red out of 10 total) and the probability of then getting red on the second (four red left out of the nine total remaining), and multiply them:

51049  =  29 probability that both marbles will be red.

However, if the question were different—for example, “What is the probability that she’ll draw one of each color”—there may be multiple sequences that get you to the desired outcome: red then blue followed by blue then red.  So calculate those sequences:

Red, then blue: five red marbles out of 10 total, and then five blue marbles out of the nine total marbles remaining:

51059  =  518

Blue, then red, is the same calculation in reverse: five blue marbles out of 10 total, and then five red marbles out of the nine marbles left:

51059  =  518

And then, since each 518 sequence is a different way to achieve the goal of one of each, add them together:

518 + 518  =  101859

More important than the results of these problems is the knowledge that “simultaneous” is just a device to make problems seem a bit more difficult when really they’re not. Simultaneous doesn’t exist, so when you’re asked to draw two things simultaneously, just calculate the probabilities of the multiple sequences that lead you to the desired outcome.

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