Question

In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is $1 third$ and the probability that he copies the answer is $1 over 6.$ The probability that his answer is correct (given that he copied it) is $1 over 8.$ Find the probability that he knew the answer to the question (given that he correctly answered it).

Solution

Let E₁ be the event that the answer is guessed, E₂ be the event that the answer is copied and E₃ the event that the examinee knows the answer.

therefore space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 1 third comma space space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 1 over 6 comma space space straight P left parenthesis straight E subscript 3 right parenthesis space equals space 1 minus 1 third minus 1 over 6 space equals fraction numerator 6 minus 2 minus 1 over denominator 6 end fraction space equals space 1 half

Let A be the event that the examinee answers correctly
We assume that only one of the multiple answers is correct.

straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis space equals space Probability space of space answering space correctly space by space guessing space equals 1 fourth straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis space equals space Probability space of space answering space correct space by space copying space space equals space 1 over 8 straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis space equals space Probability space of space answering space correctly space by space knowing space equals space 1

therefore space space straight P left parenthesis straight E subscript 3 divided by straight A right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 3 right parenthesis. space space straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis. space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis. space straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis space plus space straight P left parenthesis straight E subscript 3 right parenthesis. space straight P left parenthesis straight A divided by straight E subscript 3 right parenthesis end fraction

equals space fraction numerator begin display style 1 half end style cross times 1 over denominator begin display style 1 third end style cross times begin display style 1 fourth end style plus begin display style 1 over 6 end style cross times begin display style 1 over 8 end style plus begin display style 1 half end style cross times 1 end fraction space equals space fraction numerator begin display style 1 half end style over denominator begin display style 1 over 12 end style plus begin display style 1 over 48 end style plus begin display style 1 half end style end fraction equals space fraction numerator begin display style 1 half end style over denominator begin display style fraction numerator 4 plus 1 plus 24 over denominator 48 end fraction end style end fraction space equals 1 half cross times 48 over 29 space equals 24 over 29.

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