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Math Help - True false question probability

  1. #1
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    Post True false question probability

    Hello to all,
    I have an question that "In 20 questions of true false student require 8 right answer to pass to passout. What is the probability that student will fails?"
    Please help me to solve this.
    Thnks in advance.
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  2. #2
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    Hello, rax012!

    Youe grammar is strange . . .


    In a test of 20 true-false questions, a student needs at least 8 right answers to pass.
    What is the probability that student will fail?
    I will assume that the student randomly guesses the answers.

    To fail, he must have less than 8 right answers.
    There are 8 cases to consider . . .

    . . P(\text{0 right}) \:=\:{20\choose0}\left(\frac{1}{2}\right)^0\left(\  frac{1}{2}\right)^{20}

    . . P(\text{1 right}) \:=\:{20\choose1}\left(\frac{1}{2}\right)^1\left(\  frac{1}{2}\right)^{19}

    . . P(\text{2 right}) \:=\:{20\choose2}\left(\frac{1}{2}\right)^2\left(\  frac{1}{2}\right)^{18}

    . . P(\text{3 right}) \:=\:{20\choose2}\left(\frac{1}{2}\right)^3\left(\  frac{1}{2}\right)^{17}

    . . P(\text{4 right}) \:=\:{20\choose2}\left(\frac{1}{2}\right)^4\left(\  frac{1}{2}\right)^{16}

    . . P(\text{5 right}) \:=\:{20\choose2}\left(\frac{1}{2}\right)^5\left(\  frac{1}{2}\right)^{15}

    . . P(\text{6 right}) \:=\:{20\choose2}\left(\frac{1}{2}\right)^6\left(\  frac{1}{2}\right)^{14}

    . . P(\text{7 right}) \:=\:{20\choose2}\left(\frac{1}{2}\right)^7\left(\  frac{1}{2}\right)^{13}


    P(\text{7 or less right}) \:=\:\bigg[{20\choose0} + {20\choose1} + {20\choose2} + {20\choose3} + {20\choose4} + {20\choose5} + {20\choose6} + {20\choose7}\bigg] \cdot\left(\frac{1}{2}\right)^{20}

    . . . . . . . . . . . =\; \bigg[1 + 20 + 190 + 1140 + 4845 + 15,\!504 + 38,\!760\bigg]\cdot \frac{1}{1,\!048,\!576}

    . . . . . . . . . . . =\;\frac{137,\!980}{1,\!048,\!576} \;=\;\frac{34,\!495}{262,\!144}

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