In answering a question on a multiple-choice test, a student either knows the correct answer or guesses it. Let 0.7 be the probability that the student knows the answer, and 0.3 be the probability that the student is guessing. Assume that a student who guesses the answer will be correct with probability 0.2. What is the probability that a student knows the answer to a question, given that she answered it correctly?

Question

10-10-2019
Mathematics

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In answering a question on a multiple-choice test, a student either knows the correct answer or guesses it. Let 0.7 be the probability that the student knows the answer, and 0.3 be the probability that the student is guessing. Assume that a student who guesses the answer will be correct with probability 0.2. What is the probability that a student knows the answer to a question, given that she answered it correctly?

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Answer:

The probability is 0.9211

Step-by-step explanation:

Let's call K the event that the student know the answer, G the event that the student guess the answer and C the event that the answer is correct.

So, the probability P(K/C) that a student knows the answer to a question, given that she answered it correctly is:

P(K/C)=P(K∩C)/P(C)

Where P(C) = P(K∩C) + P(G∩C)

Then, the probability P(K∩C) that the student know the answer and it is correct is:

P(K∩C) = 0.7

On the other hand, the probability P(G∩C) that the student guess the answer and it is correct is:

P(G∩C) = 0.3*0.2 = 0.06

Because, 0.3 is the probability that the student guess the answer and 0.2 is the probability that the answer is correct given that the student guess the answer.

Therefore, The probability P(C) that the answer is correct is:

P(C) = 0.7 + 0.06 = 0.76

Finally, P(K/C) is:

P(K/C) = 0.7/0.76 = 0.9211