Question

Consider the quadratic polynomial ax 2 bx c. suppose the coefficients are given by rolling a fair die three times. what is the probability that the roots of the polynomial are equal?

Answer 1

Complete the square to write

ax² + bx + c = a (x + b/(2a))² - (b² - 4ac)/(4a)

The quadratic has a double root at x = -b/(2a) if the constant term (b² - 4ac)/(4a) vanishes, which happens if b² - 4ac = 0.

Let A, B, and C be random variables representing the value of the first, second, and third roll of the die. They are independent and identically distributed with PMF

Pr [X = x] = 1/6

if x ∈ {1, 2, 3, 4, 5, 6}, and 0 otherwise.

We want to find

Pr [B² - 4AC = 0]

Note that

B² - 4AC = 0 ⇒ B² = 4AC ⇒ (B/2)² = AC

which tells us that B must be even, and also that AC is a perfect square. Then there are 5 possible outcomes that satisfy the conditions:

• If B = 2, then AC = 1 ⇒ A = C = 1

• If B = 4, then AC = 4 ⇒ A = 1, C = 4 or A = C = 2 or A = 4, C = 1

• If B = 6, then AC = 9 ⇒ A = C = 3

and there are 6³ = 216 total possible outcomes in the sample space. So,

Pr [B² - 4AC = 0] = 5/216