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The simple version of the answer is: each question has a 1/4 chance of getting right, and since they're independent and you can mark two answers you have 2/4 or 1/2 of getting each correct, which gives you a combined chance of 25% for the entire test. The correct analysis is the combination of chances of:
First time you picked a wrong answer on both (3/4 * 3/4) and second time you eliminated one answer from each and picked the correct one (1/3 * 1/3): 6.25%
First time you picked both right, so didn't need the second time: 6.25%
First time you picked the first one right, but the second one wrong (1/4 * 3/4) and second time you picked the correct one on the second one (1/3): 6.25%
Same as above but for the second question: 6.25%
Which is also 25% btw, the other analysis is also correct, it's just an alternate problem with the same chances as this one.
Edit: sorry, didn't read the part about getting one question right would be a passing grade, so that's easier, to get a non passing grade you need to mark wrong both questions the first time (3/4 * 3/4) and mark both wrong the second time around (2/3 * 2/3) any other combination provides at least one correct answer, this has a 25% chance, so you have a 75% chance of getting at least one question right.
But you are assuming that he cannot remember his answers from the first try, and whether they were right or wrong.
his edit does not assume that; it's the cleanest way of doing the problem
I'am considering that, which is why I subtracted one from the number of possibilities in the second try.
But in order to get 100% in your second try, you can have either 0% or 50% in the first.
An unknown factor is if you even get to make a second try at getting 100% if you already passed with 50% on the first test. If it is possible to redo a passed test, I still find it unlikely that anyone would do so given that they know that they don't know the answers.
Including the edit that you're not told which one was right in the first attempt with a 50% score, it makes a lot more sense to accept the first 50% pass. Choosing different answers for the second try would only give the maximum score of 50% again, while choosing completely random answers again would only give the same chance as the first attempt, in which 0% is still more likely than 100%
Similarly, if you do get 100% on the first attempt, why'd you want to try again.. a lot of the answers here calculate the overall statistics when using both attempts regardless.
Yes, I took that into consideration, those are my scenarios 1 (0% on the first try), and 3 and 4 (both with 50% on the first try). Scenario 2 has 100% in the first try, thus accounting for all the possible ways to get to 100% in up to two tries.