Ask Lemmy

The probability of getting both right the first time is easy: 0.25*0.25 = 0.625 or 6.25%

The probability of getting exactly one right is: either you get the 1st one right and miss the second, or vice versa. Thats 0.25*0.75 + 0.75*0.25 = 0.5*0.75 = 0.375 or 37.5%, so the probability of getting at least 50% is 0.375 + 0.0625 = 0.4375 or 43.75%, even without retries, so pretty good odds. The probability of missing both is 1 - 0.4375 = 0.5625 (or 0.75*0.75).

When you retry, there's two possibilities:

1. You missed both: now your probability of getting at least one of them right is: (1/3)(1/3) + 2*(1/3)(2/3) =~ 55.55%
2. You got only one wrong: you just need to guess the other, so it's 100% for you to get at least one, and 1/3 (33.33%) to get both

So, including a retry, you either:

1. Guess them both the first try: 0.0625 or 6.25%
2. Guess one of them, then guess both: 0.375*(1/3) = 0.125
3. Guess one of them, then still guess only one: 0.375*(2/3) = 0.25 or 25%
4. Guess none first, then guess one: 0.5625*2*(1/3)(2/3) = 0.25 or 25%
5. Guess none first, then guess both: 0.5625*(1/3)*(1/3) = 0.0625 or 6.25%
6. Guess none, then still guess none: 0.5625*(2/3)*(2/3) = 0.25 or 25%

So, probability of a passing grade is 75%. Not a very good test if it's so easy to pass by random guessing ;)