2011 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2011 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 12B solutions, or check the answer key.

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Concepts:geometric probability independent events

Difficulty rating: 1390

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?

$\dfrac{1}{9}$

$\dfrac{1}{3}$

$\dfrac{4}{9}$

$\dfrac{5}{9}$

$\dfrac{2}{3}$

Solution:

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.

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