Each coefficient in equation $ax^2 + bx + c = 0$ is determined by throwing an ordinary die. What is the probability that the equation will have real roots?
How do I go about doing this? Any help is much appreciated thanks.
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4Please avoid dropping your questions on us. What have you tried? where did you get stuck? – Ittay Weiss Mar 08 '16 at 10:51
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1What do you mean by an ordinary die? so the numbers $a,b$ and $c$ can only take values between $1$ and $6$? – Ove Ahlman Mar 08 '16 at 10:51
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@OveAhlman i think so. – Taylor Ted Mar 08 '16 at 10:54
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The solution of a quadratic equation is given by $x = \frac{-b +/- \sqrt{b^2 - 4ac}}{2a}$ so as long as the determinant $b^2 - 4ac$ is positive, there will be a solution to the equation. All that is left is to count the possibilities. – Horus Mar 08 '16 at 10:57
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To fnd the roots there is a formula with a square root. This square root must be greater or equal than zero. – cgiovanardi Mar 08 '16 at 10:59
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Another dup candidate – ccorn Mar 08 '16 at 11:11
1 Answers
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To have solutions, the discriminant $b^2-4ac$ must be positive. To count how many possibilities of $216$ total there are such that $b^2-4ac$ is not positive, we can just try them all. A simple Mathematica script:
r = 0;
For[a = 1, a <= 6, a++,
For[b = 1, b <= 6, b++,
For[c = 1, c <= 6, c++,
If[b^2 - 4*a*c < 0, r++];
];
];
];
r/(6^3)
Results in $\frac{173}{216}$, which is about $0.8009\ldots$ (that's the chance that the equation is not solvable).
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1Playing with icosahedrons instead of cubes, we would get $\frac{3067}{4000}=0.76675$ ! Using disdyakis triacontahedrons $\frac{215993}{288000}\approx 0.75000$. Funny ! – Claude Leibovici Mar 08 '16 at 11:35