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Math Help - If this is true, what is the probability.

  1. #1
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    If this is true, what is the probability.

    Question:
    Suppose a student is about to take a multiple choice test has only learned 60% of the material covered in the exam. thus, there is a 60% chance that she will know the answer to the question. However, if she does not know the answer to a question, she still has a 20% chance of getting the right answer by guessing.
    a) if we choose a question at random from the exam, what is the probability that she will get it right?
    20% of 40 = 8
    60% + 8% = 68% chance of getting it right

    b) If we know that she correctly answered a question in the exam, what is the probability that she learned the material in the question.

    this is where I have trouble, my thoughts were to do it the following. Because we are told that she is correct we are only concerned about when she gets a question right (part a), so.

    60/68 * 100 = 88.24%

    It has been suggested that this is not the right way to go about it, instead.

    Let C be the event "she got it right", G be the event "she guessed correctly" and K be the event "she knows the material".
    Since the question states she got it right, P(C)=1
    The probability for guessing an answer correctly is P(G)=0.2
    So, if she got it right, P(K)=P(C)-P(G)
    thus, P(K)=1-0.2
    P(K)=0.8


    Can someone show me which method is correct and more importantly, why the other method is incorrect. thanks for you time and assistance.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: If this is true, what is the probability.

    The answer 88.24% is correct. This is a conditional probability question, which can be solved using Bayes Theorem:

    The probability that she knew the material given that she got the answer right = P(she knows | she got it right) = P(she knows AND got it right)/P(She got it right) = 0.6/0.68 = 0.8824.

    Another way to think it thorugh is like this: there are 68 correct answer ouit of 100, and 60 of those are becuase she knew the material, so 60/68 = 0.8824.

    Your other approach is incorrect because P(K) does not equal P(C)-P(G) as you've defined them. P(G) should properly be defined as the probability that she guessed given that she got it right, which is 8/68. So then your approach becomes P(K|C) = P(C) - P(G|C) = 1-0.1176 = 0.8824.
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