# Simple probability question I can't solve: not enough data?

The pupil knows how to translate the word "art" from English to Belarusian. That's true, teacher knows it. In a year, the teacher wants to check if the pupil still remember the translation of the word. So, the teacher gives him a multiple choice quiz to answer this question. There are 5 options to answer, only one is correct. The correct answer was A and the pupil decided to choose A as an answer (anyway he should choose any answer among 5 options).

So, what is the probability the pupil REALLY remembers how to translate "memory"?

My guessing... Let me started, please.

Lets say the event the pupil remembers how to translate the word is R. We have to find P(R) - the probability the pupil still remembers the translation.

Let's A is the event the person chooses A in the quiz.

We have two conditional probabilities:

P (A|not R)=0.2 (he doesn't know the answer)

P (A|R)=1 (if he knows how to translate, he answers A, because that is the correct answer).

We have to find P(R).

Can we or we can't?

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If you're willing to specify your prior belief about the probability that the student remembers, then you can use Bayes' Formula to update and obtain the posterior probability that he remembers given that he answered correctly. Otherwise, as kaine notes, you cannot reach any conclusion. –  Jonathan Christensen Dec 20 '12 at 22:09

If the only alternatives are $R$ (he remembers the right word) and $R^c$ (he forgets the word completely and just takes a random guess), then you can say that, if $A$ is the event of getting the correct answer, $$P(A) = P(A|R) P(R) + P(A|R^c) P(R^c) = P(R) + \frac{1}{5} (1 - P(R)) = \frac{1}{5} + \frac{4}{5} P(R)$$ so that $$P(R) = \frac{5 P(A) - 1}{4}$$ Thus if you gave a test with this type of question for $n$ different words, and the student got $x$ answers correct, an unbiased estimator for the fraction of those $n$ words that the student remembered would be $(5 x/n - 1)/4$.