Question

In answering a question on a multiple choice test, a student either knows the answer or guesses. Let $3 over 4$ be the probability that he knows the answer and $1 fourth$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $1 fourth.$ What is the probability that the student knows the answer given that he answered it correctly?

Solution

Let E₁, E₂, E be the events
E₁ : ‘the student knows the answer’
E₂ : ‘the student guesses the answer’,
E : ‘the answer is correct’
   $therefore space space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 3 over 4 space space and space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 1 fourth$
$straight P left parenthesis straight E space vertical line space straight E subscript 1 right parenthesis space equals space 1 space space space and space straight P left parenthesis straight E space vertical line space straight E subscript 2 right parenthesis space equals 1 fourth$
   $open square brackets because space space when space the space student space knows space the space answer comma space it space is space straight a space sure space event close square brackets$
$Required space probability space equals space straight P left parenthesis straight E subscript 1 vertical line straight E right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight E vertical line straight E subscript 1 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight E space vertical line space straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight E space vertical line space straight E subscript 2 right parenthesis end fraction$
(By Baye's Theorem)
   $equals space fraction numerator begin display style 3 over 4 end style cross times 1 over denominator begin display style 3 over 4 end style cross times 1 plus begin display style 1 fourth end style cross times begin display style 1 fourth end style end fraction equals space fraction numerator begin display style 3 over 4 end style over denominator begin display style 3 over 4 end style plus begin display style 1 over 16 end style end fraction equals fraction numerator begin display style 3 over 4 end style over denominator begin display style fraction numerator 12 plus 1 over denominator 16 end fraction end style end fraction equals space 3 over 4 cross times 16 over 13 space equals 12 over 13.$

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