2011 AMC 10B Problem 13

Problem 13 of 25IntermediateCounting & Probability

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?

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Solution

A selected number is negative with probability

\frac{20}{30}=\frac{2}{3}

and positive with probability

\frac{10}{30}=\frac{1}{3}.

(Selecting exactly

0

has probability

0.

) The product is positive exactly when both numbers have the same sign, so the probability is

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

Thus, the correct answer is D .