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Solve the Following Question and Mark the Best Possible Option. in a Test, an Examinee Either Guesses Or Copies Or Knows the Answers to a Multiple-choice Question with Four Choices. the Probability - Mathematics

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MCQ

Solve the following question and mark the best possible option.
In a test, an examinee either guesses or copies or knows the answers to a multiple-choice question with four choices. The probability that he makes a guess is `1/3` and the probability that he copies the answer is `1/6`. The probability that his answer is correct given that he copied it, is `1/8`. Find the probability that he knew the answer to the question given that he correctly answered it.

Options

  • `13/27`

  • `24/29`

  • `18/31`

  • `17/31`

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Solution

Let E1 be the event that the answer is guessed E2 be the event that the answer is copied, Ebe the event that the examinee knows the answer and E be the event that the examinee answered correctly.
Given P(E1) = `1/3, P(E_2) = 1/6`

Assume that events E1, E2 and E3 are exhaustive.
P(E1) + P(E2) + P(E3) = 1
P(E3) = 1 - P(E1) - P(E2) = 1 - `1/3 - 1/6 = 1/2`

Now P(E/E1) = Probability of getting a correct answer by guessing = `1/4`
[Since these are 4 alternatives]
P(E/E2) = Probability of answering correctly by copying = `1/8`
P(E/E3) = Probability of answering correctly by knowing = 1
Clearly (E3/E) is the event he knew the answer to the question given that he correctly answered it.
∴ P(E3/E) = `[P(E_3) P(E/E_3)]/[P(E_1) P(E/E_1) + P(E_2) P(E/E_2) + P(E_3) P(E/E_3)]`

∴ P(E3/E) = `(1/2 xx 1)/(1/3 xx 1/4 + 1/6 xx 1/8 + 1/2 xx 1) = 24/29.`

Concept: Probability (Entrance Exam)
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