Competition · AMC preparation · step 4 of 4

AMC 10 · 2011B · #13

Grade 7 probability
Archetype Geometric Probability by Area └ Sub-archetype Uniform Draws to Favorable Region
geometric-probabilityprobability-basicsign-analysis casework ↑ Prerequisites: probability-basic
📏 Medium solution 💡 3 insights
Problem
Two numbers are picked at random, each anywhere in the interval [−20,10][-20, 10], and the picks do not affect each other. Find how likely it is that the two numbers multiply to something greater than zero.

Pick an answer.

(A)
$\frac{1}{9}$
(B)
$\frac{1}{3}$
(C)
$\frac{4}{9}$
(D)
$\frac{5}{9}$
(E)
$\frac{2}{3}$

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The key reframe (Tool #16) is that “product greater than zero” just means “both numbers have the same sign,” which splits cleanly into two separate jobs (Tool #7): both numbers negative, or both numbers positive. Each job is a small independent-probability calculation, and the two jobs never overlap, so their chances add. Tool #1 (Draw a Diagram) backs this up: picturing the interval as a number line makes the negative part (length 20) and positive part (length 10) easy to measure against the whole length 30.

1STEP 1

Measure the negative and positive parts

Total length 30: 20 below zero, 10 above. So one pick is negative with chance 23\frac{2}{3}, positive with chance 13\frac{1}{3}.

P(neg) = 20/30 = 2/3, P(pos) = 10/30 = 1/3
2STEP 2

Chance both picks are negative

First way to a positive product: both picks negative. They are independent, so multiply — 23×23=\frac{2}{3}\times\frac{2}{3}= 49\frac{4}{9}.

P(both neg) = 2/3×2/3 = 4/9
3STEP 3

Chance both picks are positive

The other way: both picks positive. Multiply again — 13×13=\frac{1}{3}\times\frac{1}{3}= 19\frac{1}{9}.

P(both pos) = 1/3×1/3 = 1/9
4STEP 4

Add the two separate cases

The two cases never overlap, so add them: 49+19=\frac{4}{9}+\frac{1}{9}= 59\frac{5}{9}, which is choice (D).

P(product ＞ 0) = 4/9 + 1/9 = 5/9 → (D)
Answer
5/9
Check with the complement. The product is negative when the two picks have opposite signs, which happens two ways: neg-then-pos or pos-then-neg, giving 2×2/3×1/3=4/9. Since the chance of landing exactly on 0 is 0, positive and negative products must fill the whole probability: 5/9+4/9=1. It checks out, and 5/9 is more than half, which makes sense because the fatter negative side raises the chance of a same-sign match.
💡Key takeaway

A product is positive only when both numbers share a sign, so find each same-sign chance and add them up.

  • Measure the negative and positive parts
  • Chance both picks are negative
  • Chance both picks are positive
  • Add the two separate cases

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