How to Calculate the Probability of Getting Four or More Correct Answers on a Multiple-Choice Exam by Guessing

Taking a multiple-choice exam can be nerve-wracking, especially if you're not sure about the correct answer. But what are the chances of getting four or more correct answers just by guessing? In this blog post, we'll explore how to calculate the probability of this happening.

On a multiple-choice exam with three possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing?

Let's start with the basics. If there are three possible answers for each of the five questions, then there are 3^5 = 243 possible ways to answer all five questions. To calculate the probability of getting four or more correct answers, we need to calculate the number of ways to get four, or five, correct answers and divide that by 243.

To get four correct answers, we can choose four questions to answer correctly out of the five, and then we can choose any of the three answers for those four questions. The remaining question can be answered incorrectly in any of the three ways. So, the number of ways to get four correct answers is:

5C4 * 3^4 * 3^1 = 405

where 5C4 is the number of ways to choose four questions out of five, and 3^4 and 3^1 are the number of ways to choose the correct and incorrect answers for those questions, respectively.

Similarly, to get five correct answers, we can choose all five questions to answer correctly, and we can choose any of the three answers for each of those questions. So, the number of ways to get five correct answers is:

1 * 3^5 = 243

where 1 is the number of ways to choose all five questions to answer correctly.

Therefore, the total number of ways to get four or more correct answers is 405 + 243 = 648. And the probability of getting four or more correct answers is:

648/243 = 2.67

So, the probability of getting four or more correct answers just by guessing is approximately 2.67%.

Of course, this calculation assumes that the student is purely guessing, and does not have any knowledge of the correct answers. If the student has some knowledge of the correct answers, then the probability of getting four or more correct answers would be higher.

In conclusion, getting four or more correct answers on a multiple-choice exam with three possible answers for each of the five questions just by guessing is not very likely, with a probability of approximately 2.67%. It's always best to prepare for the exam by studying the material thoroughly and taking practice tests to increase your chances of success. Good luck on your next exam!