Let us consider the quadratic equation: ax˛ + bx + c = 0.
We assign values to the coefficients a, b and c by throwing a die.
What is the probability that the equation will have real roots ?
A six-sided die tossed randomly to determine the coefficients for a, b, and c of a quadratic equation can yield 6 x 6 x 6 = 216 unique equations. We want to determine those that have real roots. We can do so by eliminating those with imaginary roots. We need to find all values of a,b and c such that 4AC is greater than B^2. For a = b = 1, there are 36 combinations of 4AC that are greater than 1 squared. For a b of 2, there are 35 values of AC such that 4AC >4. For an b of 3, there are 33 values of AC such that 4AC exceeds 9. For a B of 4 the total is 28, for 5 the total is 22 and for 6 the total is 19. Thus we have 173/216, or 80.1% probability of irrational roots.
Gordon Steel
Die Based Coefficients
