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!

  • @ilmagico
    5•11 months ago

    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 ;)