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I was doing some AP Biology practice, and noticed that instead of Bio, I was learning more about probability. Tests have A B C and D, say you could guess one, get an answer whether it was right or wrong, guess again, and again and again, until eventually you got it right.

For guessing the first one right, the probability is just $\frac{1}{4}$, guessing the first wrong and second right is $\frac{3}{4} \cdot \frac{1}{3} = \frac{1}{4}$, the first two wrong and third right, $\frac{3}{4} \cdot \frac{2}{3} \cdot \frac{1}{2} = \frac{1}{4}$, and in the case of all everything wrong, the same value, as it is the same equation. Its just interesting, that all these values are the same, even though intuitively it seems like getting the first one right or getting them all wrong would be the more likely one, just wondering if there is a good explanation for why this happens, as it really surprised me to actually look at the numbers.

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  • $\begingroup$ Intuitively, in order to get all the answers wrong you have to know which is the right answer. So you would expect that guessing them all wrong would have the same probability as guessing them all right. $\endgroup$
    – postmortes
    Jul 28, 2020 at 7:43

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Let us say that even after guessing the correct answer, we keep on guessing until we have guessed all possible answers. Obviously, this does not alter the probability.

The question become “what is the probability that the correct answer is the first, second, third, fourth guess?”. The answer is of course $\frac{1}{4}$

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