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Math Help - probability

  1. #1
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    probability

    I need help with my homeworks . Maybe somebody knows how to solve these 3 tasks.
    P.S. Thank you very much

    1. A student studying for a vocabulary test knows the meanings of 12 words from a list of 20 words. If the test contains 10 words from the study list, what is the probability that at least 8 words on the test are words that tha student knows?

    2. A biology quiz consist of 8 multiple-choice questions. 5 must be answered correctly to recieve a passing grade. If each questions has 5 possible answers, of which one is correct.What is the probability that a student who guesses at random on each question will pass the examination?

    3. A panel of 64 economists was asked to predict the average unemployment rate for the upcoming year. The results of survey follow:

    Unemployment rate,% 4.5 | 4.6 | 4.7 | 4.8 | 4.9 | 5.0 | 5.1 |
    Economists''''''''''''''''''''''''''''' 2 | 4 | 8# | 20 #| 14 | 12 | 4 |

    Based on this survey, what does the panel expect the average unemployment rate to be next year?
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  2. #2
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    Know 12 of 20. Want 8, 9, or 10 on list of 10.
    1.

    Ok, well the chances of knowing all 10 are
    (12/20)*(11/19)*(10/18)*(9/17)*(8/16)*(7/15)*(6/14)*(5/13)*(4/12)*(3/11)

    The chances
    of knowing 9 of the 10 are
    (12/20)*(11/19)*(10/18)*(9/17)*(8/16)*(7/15)*(6/14)*(5/13)*(4/12)*(the chance of not knowing the 10th question), where the chance of not knowing the 10th question is (8/11). Then you have to take into account that this can happen in 10 different ways - because there are 10 different choices for the question that the student gets wrong - (ie. it could be the first OR the second OR the third.. etc...) So you multiply your answer by 10.

    The chances of knowing 8 of 10 are similarly constructed:
    (12/20)*(11/19)*(10/18)*(9/17)*(8/16)*(7/15)*(6/14)*(5/13) * (chances of not knowing the last 2), where the chances of not knowing the last two are (8/12)*(7/11).
    Now the two questions answered wrongly can be any of the 10, and you use your 10C2 button on your calculator to get this. (incidentally - you could have done this in the previous part - 10C1 = 10, which is what we got).

    So now you have the chances of getting exactly 8, exactly 9 or exactly 10.
    So in order to find the chances of getting AT LEAST 8, you just add up the previous three answers.
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  3. #3
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    Quote Originally Posted by aura000 View Post

    2. A biology quiz consist of 8 multiple-choice questions. 5 must be answered correctly to recieve a passing grade. If each questions has 5 possible answers, of which one is correct.What is the probability that a student who guesses at random on each question will pass the examination?
    --------------------------------------------------------------------------

    You need at least 5 correct answers.
    That is either 5, 6, 7 or 8 correct answers.

    The chances of randomly guessing all 8 correctly are:
    (1/5)^8.
    Or to be consistent with the method used for the following parts:
    (1/5)^8 * (8C0)


    For 7:
    (1/5)^7 * (4/5) * (8C1)

    For 6:
    (1/5)^6 * (4/5)^2 * (8C2)

    For 5:
    (1/5)^5 * (4/5)^3 * (8C3)

    Add these four answers.
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  4. #4
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    Quote Originally Posted by aura000 View Post
    3. A panel of 64 economists was asked to predict the average unemployment rate for the upcoming year. The results of survey follow:

    Unemployment rate,% 4.5 | 4.6 | 4.7 | 4.8 | 4.9 | 5.0 | 5.1 |
    Economists''''''''''''''''''''''''''''' 2 | 4 | 8# | 20 #| 14 | 12 | 4 |

    Based on this survey, what does the panel expect the average unemployment rate to be next year?
    --------------------------------------------------------------------------
    If you think of the number of economists as the weight of each rate, then you just calculate the average as per usual:
    (4.5*2 + 4.6*4 + 4.7*8 + 4.8*20 + 4.9*14 + 5.0*12 + 5.1*4)/64.
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