2012 AMC 12B Problem 13

Question

Two parabolas have equations

y = x^2 + ax + b

and

y = x^2 + cx + d,

where

a,

b,

c,

and

d

are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common?

\dfrac{1}{2}

\dfrac{25}{36}

\dfrac{5}{6}

\dfrac{31}{36}

1

Answer: D

Concepts:basic probability complementary counting dice (probability)

Difficulty rating: 1590

Small hint:

Setting the two right sides equal, the

x^2

terms cancel, leaving

ax+b=cx+d

Big hint:

There is no common point exactly when

a=c

and

b\neq d;

find that probability and subtract from

1

Solution:

The parabolas meet where

x^2+ax+b=x^2+cx+d,

i.e.

ax+b=cx+d.

This has no solution exactly when the lines are parallel and distinct:

a=c

and

b\neq d.

The probability that

a=c

is

\tfrac16,

and the probability that

b\neq d

is

\tfrac56,

so the probability of no common point is

\tfrac16\cdot\tfrac56=\tfrac5{36}.

The probability of at least one common point is

1-\tfrac5{36}=\tfrac{31}{36}.

Thus, the correct answer is D.