AMC 10 · 2017 · #15

Grade 7 probability

Archetype Geometric Probability by Area └ Sub-archetype Uniform Draws to Favorable Region

geometric-probabilityprobability-basic caseworksymmetry-argument ↑ Prerequisites: probability-basic

📏 Medium solution 💡 3 insights

Problem

Chloe picks a real number uniformly at random from [0, 2017], and independently Laurent picks a real number uniformly at random from [0, 4034]. Find the probability that Laurent's number is larger than Chloe's number.

Pick an answer.

(A)

$\frac{1}{2}$

(B)

$\frac{2}{3}$

(C)

$\frac{3}{4}$

(D)

$\frac{5}{6}$

(E)

$\frac{7}{8}$

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Two numbers picked at random become a single point whose two coordinates are those numbers, so Tool #1 (Draw a Diagram) turns the whole question into one picture: a rectangle holding every possible outcome. Because the picks are uniform, every spot in the rectangle is equally likely, so a probability is just an area divided by the rectangle's area. Tool #7 (Identify Subproblems) splits that rectangle along the diagonal L=C into a small "Chloe wins" triangle whose area is easy to find. Tool #16 (Count the Complement) then finishes fast: find the chance Chloe wins and subtract it from 1.

1STEP 1

Make every outcome a point in a box

Plot each outcome (C, L) as a point filling a rectangle 2017 wide and 4034 tall; any probability is its area over the total 2017 × 4034.

outcomes = {(C, L) : 0 ≤ C ≤ 2017, 0 ≤ L ≤ 4034}, total area = 2017 × 4034

2STEP 2

Cut the box along the line L = C

The line L = C splits the box; below it Chloe wins (L ＜ C), a right triangle with legs 2017, so its area is 2017²/2.

Chloe-wins triangle area = 1/2 · 2017 · 2017 = 2017²/2

3STEP 3

Chloe's winning chance is one quarter

Since 4034 = 2 × 2017, the box area is 2 · 2017², so Chloe's chance is (2017²/2)/(2 · 2017²) = 1/4 — the exact numbers cancel.

P(Chloe wins) = (2017² / 2)/(2 · 2017²) = 1/4

4STEP 4

Take the complement to finish

A tie has zero area, so Laurent wins whenever Chloe doesn't: 1 - 1/4 = 3/4, matching choice (C).

P(Laurent wins) = 1 - 1/4 = 3/4 = (C)

Answer

3/4

3/4 is more than 1/2, which makes sense: Laurent draws from a range twice as long, so half the time his number already lands in [2017, 4034] — above anything Chloe can pick — and he wins outright. A second route confirms it: with probability 1/2 Laurent's number is above 2017 (a sure win), and with probability 1/2 it falls in [0, 2017], where the two numbers are symmetric and he wins half the time, giving 1/2 · 1 + 1/2 · 1/2 = 3/4. Both agree with (C).

💡Key takeaway

When two random numbers are compared, picture them as a point in a rectangle — then the probability is just the slice of area where your condition is true.

Make every outcome a point in a box
Cut the box along the line L = C
Chloe's winning chance is one quarter
Take the complement to finish

Make every outcome a point in a box

Cut the box along the line L = C

Chloe's winning chance is one quarter

Take the complement to finish

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