Hi,
I just did a test which had two multiple choice questions. Each question was worth one point. Getting them both right would result in getting a 100% score. Suffice it to say, getting just one question right would give you 50% and with that a passing grade.
So you have two multiple choice questions. Both of which are unrelated to the other. Each question has four possible answers. When you finish the test. You get to have one more try. The questions and possible answers remain the same.
Let’s say you use both tries and you remember your previous two respected answers. What would your odds be, if you were to brute force guess your way through this test, to get a passing grade or a 100%?
Edit: Both questions only have one correct answer.
IMPORTANT EDIT: YOU DO NOT KNOW WHICH ANSWER YOU HAD RIGHT OR WRONG THE SECOND TIME AROUND. You only know how many questions you got right. But you don’t know which. Sorry for the confusion!
I see some correct solutions for the 50% case here already, so this reply is going for a perfect score within two tries.
There are 16 ways to answer the quiz, one of which is correct. Assuming you don’t repeat your previous answers, two attempts give you a 2/16 or 1/8 chance that one of them is perfect.
Now if you get feedback between your attempts, you should be able to do better. Let’s see by how much and break it into cases:
Your first guess is already perfect. This happens 1/16 of the time. No further guessing is needed.
Your first guess is 50% correct. This happens 3/8 of the time. Picking one of the unguessed answers improves your score to 100% 1/6 of the time.
Your first guess is completely wrong. This happens 9/16 of the time. Picking different answers for both questions wins 1/9 of the time.
So the overall chance of a perfect score is the weighted sum of these cases or 1/16 + (3/8 * 1/6) + (9/16 * 1/9) = 3/16.