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Math Help - Probability of at least one real root

  1. #1
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    Probability of at least one real root

    Hi, if a quadratic equation is of the form:
    x^2+sqrt(p)*x+q=0 where p can be any value in range [0,a] and q in the range [-b,b] then what is the probabillty that at least one root of the equation is real?
    E.g:
    if a = 4 and b = 2 ,then
    probability is 0.625 and for a = 1 and b =2
    probability = 0.53125
    Thanks.
    Last edited by pranay; April 19th 2011 at 08:27 AM. Reason: misspelled title
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint


    The equation has at least a real root iff p - 4 q >= 0 . The asked probability is the quotient of two adequate areas in the pq plane.
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  3. #3
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    Quote Originally Posted by FernandoRevilla View Post
    Hint


    The equation has at least a real root iff p - 4 q >= 0 . The asked probability is the quotient of two adequate areas in the pq plane.
    Thanks but i couldn't get what's quotient of two adequate areas in the pq plane?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by pranay View Post
    Thanks but i couldn't get what's quotient of two adequate areas in the pq plane?

    For example, for a = 4 , b = 2 the region of the pq plane satisfying

    0 <= p <= 2 and - 4 <= q <= 4 is a square ( S )


    The region satisfying

    0 <= p <= 2 and - 4 <= q <= 4 and p >= 4q is a trapezium ( T )

    Then, the probability is

    P = ( area of T ) / (area of S )
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