What is the probability of guessing on a multiple-choice test?

While there are a variety of question styles on Microsoft exams, the bulk of them are still multiple choice. While knowing the material is key to getting those questions right, there are still some tricks to multiple choice questions that can trip up even the brightest expert. In this article, Brien Posey offers some hints for improving your odds of answering multiple-choice questions correctly.

Over the years Microsoft has experimented with a number of different types of questions on their certification exams. Even so, multiple-choice questions seem to make up the bulk of the questions on most of the exams. That being the case, I think that it makes a lot of sense to talk about some techniques for statistically improving your odds of answering multiple-choice questions correctly.

Some Bad Advice

When I was in high school, it seemed that we were constantly being forced to take all manner of standardized tests. These tests were always multiple-choice, and there were usually four possible answers to each question. I have one teacher who told the class to choose B as the answer to any question that we did not know the answer to.

The basic idea behind this advice was that if you left a question blank then you had no chance of getting the question right. However, if you made a guess then you had a 25% chance of answering the question correctly. Statistically, the chances of answering questions correctly go down if you choose a random answer for each question. However, if you answer each unknown question in a consistent manner (such as by choosing option B for each question) then you will usually yield a higher percentage of correct answers than you would if you made a random guess.

So why do I consider this to be bad advice? It's because statistically this method only yields about 25% correct answers. While that percentage is not necessarily bad, you can usually do better if you apply some reasoning and logic.

Types of Multiple Choice Questions

Before I talk about how to maximize your chances of answering multiple-choice questions correctly, I need to point out that over the last few years Microsoft has taken measures to make it a lot more difficult for exam candidates to guess a correct answer. One of the ways that they have done this is by using several different types of multiple-choice questions.

In the old days, the multiple-choice questions were fairly straightforward. For each question there were usually four answers to choose from, and the test was designed so that you could only pick one answer for each question. These types of questions still exist on Microsoft certification exams, and they are the multiple-choice question type for which it is easiest to guess a correct answer. Therefore, most of the methods that I will be discussing focus on this type of question.

The other types of multiple-choice questions that you may encounter include:

  • Multiple choice, multiple answer: These questions require you to choose multiple correct answers. For example, a question may ask you to choose all of the answers that correctly apply to a situation. Because you must choose every correct option in order for these types of questions to be scored correctly, these questions are extremely difficult to guess at.
  • Multiple choice, best answer: These questions are relatively rare on Microsoft certification exams. You will only encounter these types of questions on Pro level exams. With a normal multiple-choice question, there is only one correct answer. With a best answer question, however, there may be several potentially correct answers but you must choose the answer that is the most correct. Microsoft defines the most correct answer as the one that would require the least effort and least expense to implement while also achieving the desired outcome and adhering to Microsoft's best practices. Keep in mind that the best answer questions are situation-specific. You must choose the option that is the best fit for the situation described, even if the solution is not necessarily the best option for every situation.
  • Multiple choice, repeated scenario: These questions are commonly referred to as case study questions. These questions tend to appear mostly on Pro level exams. The basic idea is that the exam will provide you with a long and detailed scenario (typically a case study) and then ask a series of multiple choice questions that are all based on that scenario. From a statistical guessing standpoint, repeated scenario questions can be treated the same as regular multiple choice questions.
  • Multiple choice, extended matching: These questions are relatively new to Microsoft certification exams, but they have been used on medical licensure exams for years. These questions are similar to repeated scenario questions in that there is a series of questions that are all based around a common scenario. What makes these questions different is that the list of possible answers is extended. Rather than allowing you to choose from four possible answers, an extended matching question might present ten possible answers. The same list of possible answers is presented for each question in the series. Some answers might be correct for more than one question, while other answers might not be correct for any of the questions.

Tilting the Odds in Your Favor

As I previously mentioned, your odds of correctly guessing an answer are the best for multiple-choice questions with a single correct answer. Regardless of the type of question, however, you should always guess when you don't know an answer because Microsoft does not impose a penalty for incorrect guesses. You never know when a single correct guess might be just enough to put you over the pass/fail threshold.

One of the most important things that you should do to tilt the odds in your favor is to eliminate implausible answers. For example, if a multiple-choice question has four possible answers and you are able to eliminate two of those answers then you improve your odds of guessing the correct answer from 25% to 50%.

Another thing to watch out for is answers that are completely unfamiliar to you. If you have diligently prepared for an exam, yet an answer choice seems to be completely foreign, then that choice is more than likely incorrect. Microsoft does not put nonsense answers on their certification exams, but there are instances in which an incorrect answer choice falls outside of the realm of the stated exam objectives.

Conventional wisdom often dictates that if two answers look-alike but are slightly different, one of the two choices is probably correct. However, my own personal experience with Microsoft certification exams has been that often the opposite holds true. More times than not, it seems that very similar answers tends to be distractors.

If a question presents a series of numerical answers and you are unable to illuminate any of the possible choices with any certainty, that a good rule of thumb is to throw out the lowest and the highest possible value. Even though this method does not always work, the correct answer is often a mid-range value.

Another thing that you can do to improve your odds of answering the question correctly is to look for answer choices that are complete opposites of one another. Often times one of the two answers is going to be correct.

Another great strategy is to use what I like to call the true/false test. The basic idea is that if any part of a potential answer is false then the entire answer is incorrect. Although I have used this method to narrow down potential answers on more than one occasion, it only works if you actually know the material that you are being tested on, and if the answer choices are relatively long.

Along the same lines, watch out for words that convey an absolute situation. Words such as always and never reflect that an answer choice would apply regardless of the situation. Answers containing words like always and never are typically incorrect.

One of the tricks that I like to use when I take an exam is to use clues found in the questions that I do know the answer to as a way of figuring out questions that I don't know the answer to. Microsoft works hard to make sure that none of the exam questions blatantly convey the answer to another question, but often, one question will provide hints to another question’s answer.

Every once in a while you might be able to eliminate a possible answer choice by using grammatical context. Even though Microsoft has gotten a lot better about this over the years, you might still occasionally see questions in which one or more of the answer choices simply does not fit with the grammatical context of the question. Such answers can be instantly eliminated.

Finally, if you are having trouble figuring out which answers can be eliminated, don't assume that the longest, most complicated answer choice is correct. Often at least one or two of the possible choices will be unnecessarily complicated and are designed to serve as distractors for those who do not know the material.

Let's use a simple analogy. Suppose you have a fair die numbered from $1$ to $6$ inclusive. If you roll the die once, the chance of getting a $6$ is $1/6$. If you roll it a second time, the chance of getting a $6$ on the second roll is again $1/6$--the die does not "remember" what it rolled in the past. Each time you roll the die, the outcome is independent of any previous rolls. For any given roll, the chance of getting a $6$ remains $1/6$.

However, if the goal is to keep rolling until you get a $6$, then it clearly makes sense that the more times you roll, the more likely you are to eventually get a $6$. For example, if you rolled only once, the chance is $1/6$. But if you get to roll twice, then the chance that you get $6$ at least once is $$1 - (1 - 1/6)^2 = \frac{11}{36} > \frac{1}{6}.$$ If you get to roll three times, this chance increases to $$1 - (1 - 1/6)^3 = \frac{91}{216}.$$ If you get to roll $48$ times, this chance goes all the way up to $$1 - (1 - 1/6)^{48} = \frac{22448704993675756739157855502454784191}{2245225770735455724008721112379267481 6} \approx 0.999842.$$ So the more rolls you get to take, the higher your chance of observing at least one $6$, even though for any given roll, the chance of a $6$ remains $1/6$.

The same principle applies in your question. Even though each test is an independent trial, the more trials you are allowed to make, the greater your chance of achieving success for one or more trials, even though the individual chance of success for any given trial is the same.

That said, the chance of a perfect score by random guessing on a multiple choice exam with more than a handful of questions is extremely small. If the chance of guessing correctly on any individual question is $1/5$, then for a $40$ question test, the chance of a perfect score is $(1/5)^{40} \approx 1.099511627776 \times 10^{-28}$. That is so vanishingly small that even with $48$ trials, the chance of at least one perfect score result is still negligible, only about $5.27766 \times 10^{-27}$. In fact, for just $40$ questions on the test, you'd have to take the test approximately $10^{26}$ times just to have a nontrivial chance of guessing correctly on all of the questions.

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