In life, it’s important to have a hobby or pastime that you find interesting. Sometimes, when the daily grind of work, school, family, social responsibilities, (updating Facebook) and preparing for the GMAT just seems like too much to handle, it’s good to take a step back. Diving into a hobby helps take your mind off things by pausing everything else and concentrating on something personal and somewhat intimate to you. One of my favorite diversions is watching movies and immersing myself in the fictional world created on screen. Surprisingly, this same distraction can be applicable to GMAT studying as well.
Within the confines of the GMAT, the expectations for students are well known. You will be faced with 37 math and 41 verbal questions, have to select from five multiple choice answers, and complete each section within 75 minutes. However, sometimes certain questions will set up arbitrary rules within this game. An obvious example is data sufficiency: a question type that always provides two statements and asks whether a certain question can be answered using these statements. Why are there not three statements? Or four statements? The official answer will be to standardize the questions and allow for easier preparation, but the truthful answer is something most parents have had to utter countless times: “Because I said so”.
The only reason these rules apply is because they were established by the GMAC to test logical thinking. However, other rules could have been set up and test takers would have had to adhere to them. In fact, any question can set up arbitrary rules and then require you to analyze the situation and provide insight. Within the game that is the GMAT, a sub-game is created with each new question, and some of these questions have very specific rules (GMAT Inception).
The difficulty with some of the arbitrary question-specific rules is that the situation is only applicable to the exact question, meaning that you don’t have long to acclimate to the circumstances. Usually, the question will provide rules that are indispensible to solving the query, so we must adhere to them or risk falling into a trap.
Let’s look at an example that highlights the sub-game nature of certain GMAT questions:
An exam consists of 8 true/false questions. Brian forgets to study, so he must guess blindly on each question. If any score above 70% is a passing grade, what is the probability that Brian passes?
As always, let’s begin by paraphrasing the question. A student is blindly guessing on a True/False question, and thus will likely get half the questions right by default. It is conceivable that he could get 0% or 100% as well, meaning this is likely a probability question of sorts. However it’s a probability question within a probability question. Once we have accepted the premise that this exam will take place, we can only analyze the possible results of the student taking this test (the irony of which is enormous).
Another excellent trick is to look at the answer choices for easily removable options. If Brian did not study a single line of text, then the expected value of his blind guesses is 50%. This means it is possible that he can pass this test if he gets lucky, but he is not expected to do well. As such, any probability above 50% can be eliminated. We will need to do the calculations to determine exactly which answer is correct, but we already know it cannot be D or E as they are both too high.
Picking among the next three choices, each with a different denominator and fairly close values would be tricky. Statistically speaking, this question is identical to a coin flip question, where True is Heads and False is Tails (or vice versa if you prefer). The chances of getting all 8 correct, just as 8 straight Heads, would be (½)^8 or 1/2^8 or 1/256. This would yield a result of 100% on the exam. Brian would undoubtedly be surprised by such a result, but it is possible for him to pass the test without getting every question right. Since there are 8 questions, each question is worth 1/8 of the final score or 12.5%. Thus Brian could miss 1 question and still manage an 87.5%. He could even squeak by with 2 errors, giving him a result of 75% on the test. Anything lower would put him below the failure threshold.
There are three ways to calculate the remaining options, so let’s look at a more likely scenario: the possibility of getting 7 correct answers on the test. This result could be achieved if Brian missed the first question and got the next 7 right, or missed the last question after getting the first 7 right, or any other such breakdown. Logically, you can deduce that there are 8 different spots where the error could be, and the remaining 7 spots are all correct. Thus if each combination of answers has a 1/28 possibility of occurring, we should end up with 8/28 or 23/28 (cancelling to) 20/25 or 1/32. We can also use the combination formula for selecting 7 elements out of 8 where the order doesn’t matter. The formula would be n!/k!(n-k)!, where n is the total (8) and k is the number of choices (7). This would yield 8!/1!*7!, which simplifies to 8. This means there are 8 possible choices to select 7 correct answers. The final step is to divide by the total number of possibilities, which still stands at 28. The last option is to determine the numerator with the repeating elements formula n!/t!f!, where t and f are the number of repeating True and False answers. The result will still be 8!/1!7!, so 8 possibilities out of the same 256 options.
Using the same strategies on 6 correct answers and 2 false answers, we can get 8!/2!6!, which is 8*7/2 or 28 possibilities. The denominator won’t change for any of these, so the probability of getting exactly 6 correct answers is 28/256 (a little less than 11%). While I’m on the subject, I’ll simply draw attention to the fact that picking two correct answers and six incorrect answers on a binary test such as this one will yield the same results as picking two incorrect answers and six correct answers. The nature of the exercise (and the formulas) makes it so symmetry is guaranteed. This may be helpful at some point on the GMAT or in life, so try to ensure you can shortcut some calculations in this manner.
Putting together our three results, the chances of passing this exam are 1/256 + 8/256 + 28/256. This sum gives exactly answer choice B: 37/256. Although it seems unlikely that going into an exam with absolutely no preparation could yield a 15% chance of passing, those are the rules stipulated on this question. The entire GMAT exam has fixed rules, so it’s important to know how to approach each question on the exam. Moreover, it’s also important to understand the adjunct rules on particular questions in order to correctly solve the problem. As Jigsaw would rhetorically ask in any Saw movie: “Would you like to play a game?”
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.