Rolling dice and finding the roots of quadratics

Probability Level 3

Suppose you roll three fair six-sided dice and write down the number on the first, second and third dice as the coefficients a,b,c in the polynomial ax^2+bx+c respectively.

What is the probability that this polynomial has no real roots ?

If this probability can be expressed as $\dfrac AB$ , where $A$ and $B$ are coprime positive integers, find $A+B$ .

The answer is 389.

1 solution

Jesse Li
Oct 1, 2018

In order for there to be no real solutions, b²-4ac<0 must be true.

If b=1, the probability of that being true is 1.

If b=2, the probability of that being true is $\frac{35}{36}$ .

If b=3, the probability of that being true is $\frac{33}{36}$ .

If b=4, the probability of that being true is $\frac{28}{36}$ .

If b=5, the probability of that being true is $\frac{22}{36}$ .

If b=6, the probability of that being true is $\frac{19}{36}$ .

You can add all the numbers together and multiply the result by $\frac{1}{6}$ , since the probability of b being any number is $\frac{1}{6}$ .

You get $\frac{173}{216}$ , which cannot be simplified, because 173 is prime, and 216 isn't divisible by 173.

173+216= 389

Rolling dice and finding the roots of quadratics

The answer is 389.

1 solution

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