A test is composed of six multiple choice questions where each question has 4 choices. If the answer choices for each question are equally likely, find the probability of answering more than 4 questions correctly.
To find the probability of answering more than 4 questions correctly, we need to calculate the probability of answering exactly 5 questions correctly and the probability of answering all 6 questions correctly, and then add those probabilities together.
First, let's find the probability of answering exactly 5 questions correctly. There are 6 choose 5 ways to choose which 5 questions you answer correctly. For each of these ways, the probability of getting those 5 questions correct and the remaining 1 question wrong is (1/4)^5 * (3/4)^1. So the probability of answering exactly 5 questions correctly is:
(6 choose 5) * (1/4)^5 * (3/4)^1
Next, let's find the probability of answering all 6 questions correctly. The probability of answering each question correctly is (1/4), so the probability of answering all 6 questions correctly is:
(1/4)^6
Finally, we add the two probabilities together to get the probability of answering more than 4 questions correctly:
(6 choose 5) * (1/4)^5 * (3/4)^1 + (1/4)^6