AMC 10 · 2019 · #20

Grade 7 probability

Archetype Overlap / Complement Counting └ Sub-archetype At-Least-One Complement: 1 − P(None)

probability-basicgeometric-probabilitycaseworkconditional-probabilityfraction-arithmetic caseworkidentify-subproblems ↑ Prerequisites: probability-basicgeometric-probabilityconditional-probability

📏 Medium solution 💡 4 insights

Problem

A number between 0 and 1 is built like this: flip a fair coin; on heads flip again, taking 0 for heads and 1 for tails; on tails draw the number uniformly from the interval. A second number is built independently the same way. Find the probability that the two numbers differ by more than one half.

Pick an answer.

(A)

$\frac{1}{3}$

(B)

$\frac{7}{16}$

(C)

$\frac{1}{2}$

(D)

$\frac{9}{16}$

(E)

$\frac{2}{3}$

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems): split into four cases by (branch for x) × (branch for y) — each independent and equally likely with probability 1/4. Tool #2 (Systematic List): enumerate all four cases (Discrete-Discrete, Discrete-Uniform, Uniform-Discrete, Uniform-Uniform). Tool #1 (Diagram): for the Uniform-Uniform case, draw the unit square and shade |x-y| ＞ 1/2 — two right-triangle corners. Tool #9 (Easier Problem): each sub-case is a small problem solvable by counting or by a clean area; combining them is just a weighted sum.

1STEP 1

Split into four cases

The two build methods give four cases.

P(DD) = P(DU) = P(UD) = P(UU) = 1/4

2STEP 2

Both are endpoints

They differ with probability one half.

P(· ∣ DD) = 1/2

3STEP 3

One endpoint, one uniform

This case is also one half.

P(· ∣ DU) = 1/2

4STEP 4

The mirror case

By symmetry it is the same.

P(· ∣ UD) = 1/2

5STEP 5

Both uniform

By area it is one quarter.

P(· ∣ UU) = 2 · 1/2 · 1/2 · 1/2 = 1/4

6STEP 6

Combine the four

Averaging gives seven sixteenths.

P = 1/4(1/2 + 1/2 + 1/2 + 1/4) = 7/16

Answer

7/16

7/16 ≈ 0.44 is less than 1/2. Intuitively if both x and y were uniform we'd get only 1/4, but the discrete branch boosts the chance — when x or y snaps to an endpoint 0 or 1, escaping the other by more than 1/2 is easier. So a number between 1/4 and 1/2 is exactly what we expect. Conversion check: 1/2 + 1/2 + 1/2 + 1/4 = (2+2+2+1)/4 = 7/4, and 1/4 · 7/4 = 7/16. Choice (D) 9/16 is the complementary probability P(|x-y| ≤ 1/2) trap.

💡Key takeaway

This AMC 12 problem only needs Grade 7 probability and unit-square geometric probability you already know — split into 4 equally likely cases (Discrete-Discrete, Discrete-Uniform, Uniform-Discrete, Uniform-Uniform), get conditional chances 1/2, 1/2, 1/2, 1/4, then average: 1/4(1/2+1/2+1/2+1/4) = 7/16.