AMC 10 · 2003 · #12

Grade 7 geometry-2d
Archetype Geometric Probability by Area └ Sub-archetype Uniform Draws to Favorable Region
geometric-probabilityarea-trianglescoordinate-geometry identify-subproblems ↑ Prerequisites: area-triangles
📏 Medium solution 💡 2 insights
Problem
A point (x,y) is chosen at random, evenly, from inside the rectangle whose corners are (0,0), (4,0), (4,1), and (0,1). Find the probability that the point's x-coordinate is smaller than its y-coordinate, that is, the probability that x ＜ y.

Pick an answer.

(A)
$\frac{1}{8}$
(B)
$\frac{1}{4}$
(C)
$\frac{3}{8}$
(D)
$\frac{1}{2}$
(E)
$\frac{3}{4}$

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

When a point is chosen evenly at random from a region, probability is just a ratio of areas: (area of the good part) over (area of the whole). So Tool #1 (Draw a Diagram) is the heart of the work — sketch the rectangle, draw the boundary line y=x, and shade the piece where x ＜ y. Tool #7 (Identify Subproblems) splits the job into two clean pieces: find the total area, then find the shaded area. Tool #14 (Extreme Principle) does the sharp thinking up front: since y never exceeds 1, the inequality x ＜ y forces x ＜ 1, so the entire long right end of the rectangle (from x=1 to x=4) is dead — the good region is trapped in the first unit of width.

1STEP 1

Total area is the sample space

Uniform choice makes probability a share of area, and the rectangle is 4 wide by 1 tall, so the denominator is 4.

total area = 4 × 1 = 4
2STEP 2

Locate the region where x ＜ y

Above the line y=x is the winning side, and since y never passes 1 the region is just the triangle (0,0), (0,1), (1,1).

y=x meets y=1 at (1,1); region = △(0,0),(0,1),(1,1)
3STEP 3

Area of the winning triangle

That triangle is right-angled at (0,1) with both legs of length 1, so its area is 1/2 — the numerator.

area = 1/2 × 1 × 1 = 1/2
4STEP 4

Divide the areas to get the probability

Divide the areas: 1/2 divided by 4 is 1/8, which is choice (A).

P(x ＜ y) = 1/2/4 = 1/8 → (A)
Answer
1/8
The winning triangle is tiny compared to the long rectangle: it fits inside the first 1 × 1 square on the left, and even there it is only half of that square. One unit square is 1/4 of the whole rectangle, and half of that is 1/8 — a small probability, which fits the picture of a skinny wide rectangle where x is usually far bigger than the at-most-1 value of y. So the small answer 1/8 is believable, and the larger choices like 1/2 or 3/4 would be far too big.
💡Key takeaway

When a point is dropped evenly on a region, the probability is just the good area divided by the whole area — here a 1/2 triangle inside an area-4 rectangle gives 1/8.

  • Total area is the sample space
  • Locate the region where x ＜ y
  • Area of the winning triangle
  • Divide the areas to get the probability

More like this

Same archetype — closest grade level first.