2011 AMC 12B Problem 9

Question

Two real numbers are selected independently at random from the interval

[-20, 10].

What is the probability that the product of those numbers is greater than zero?

\dfrac{1}{9}

\dfrac{1}{3}

\dfrac{4}{9}

\dfrac{5}{9}

\dfrac{2}{3}

The product is positive exactly when both numbers share the same sign

Each number is negative with probability

\dfrac{20}{30}

and positive with probability

\dfrac{10}{30}

The interval has length

30,

with

20

of it negative and

10

of it positive. So each number is positive with probability

\dfrac13

and negative with probability

\dfrac23.

The product is positive when both are positive or both are negative:

\left(\dfrac13\right)^2+\left(\dfrac23\right)^2=\dfrac19+\dfrac49=\dfrac59.

Thus, the correct answer is D.

Problem 9 in other years