Topics / Counting & Probability

Geometric Probability

Probability as a ratio of lengths, areas, or volumes

Geometric ProbabilityAMC 10 Step by Step

Name: Date: September 13, 2026

Counting & Probability · 16 problems have this as their main idea

primary-topic problems (1.2% of all)

more as a secondary topic

Where it appears

P1-10

P11-15

P16-20

P21-25

## What you need to know
- Continuous uniform probability: for a point chosen uniformly from a region $R$ , $P(\text{point} \in S) = \dfrac{\text{measure}(S)}{\text{measure}(R)}$ , where measure is length, area, or volume.
- Two independent uniform numbers $x, y \in [0, L]$ are one uniform point in the square $[0, L]^2$ ; conditions like $|x - y| < d$ become regions bounded by lines.
- Standard regions: $|x - y| \le d$ inside an $L \times L$ square has area $L^2 - (L - d)^2$ ; $x + y \le c$ with $c \le L$ is a triangle of area $c^2/2$ .
- A dropped coin of radius $r$ misses the lines of a tiling exactly when its center lies in the tile shrunk by $r$ on every side.

## How AMC 10 tests it
- Problems 10–18: two people arrive at random times in an hour and each waits $15$ minutes; probability they meet (the square with a diagonal band).
- Problems 12–20: a random point in a square, circle, or triangle; probability it is closer to one side, or inside a related shape.
- Problems 15–22: a random point splits a segment; probability the pieces form a triangle.
- Problems 20–24: a coin dropped on a tiled floor, probability of not touching a line.

## Standard approaches
1. Identify the random variables and their ranges; that fixes the total region.
2. Translate the event into inequalities and sketch the favorable region.
3. Compute the area by decomposition or complement.
4. For dropped objects, shrink the allowed region by the radius and take the ratio of areas.
5. Check extremes: the answer should tend to $1$ or $0$ when the parameters do.

## Worked example
Real numbers $x$ and $y$ are chosen independently and uniformly at random from $[0, 6]$ . What is the probability that $|x - y| \le 2$ and $x + y \ge 4$ ?

(A) $\frac{1}{3}$ (B) $\frac{7}{18}$ (C) $\frac{4}{9}$ (D) $\frac{1}{2}$ (E) $\frac{5}{9}$

Solution. The sample space is the $6 \times 6$ square, area $36$ . The band $|x - y| \le 2$ has area $36 - 2 \cdot \frac{1}{2} \cdot 4^2 = 20$ . Remove the part of the band with $x + y < 4$ . That corner triangle, vertices $(0,0), (4,0), (0,4)$ , has area $8$ , but its two tips outside the band, the triangle $(2,0), (4,0), (3,1)$ and its mirror image, have area $1$ each. So the band loses $6$ , leaving $14$ . Probability $\frac{14}{36} = \frac{7}{18}$ . Answer $\boxed{\textbf{(B)}\ \frac{7}{18}}$ .

## Pitfalls
- Using lengths when the event depends on two independent variables; it must be an area in the square.
- Forgetting to square when scaling: shrinking the side by a factor $k$ scales area by $k^2$ .
- Drawing the band $|x - y| \le d$ with corner triangles of leg $d$ instead of $L - d$ .
- Shrinking the tile by the coin's diameter instead of its radius.

Traps that recur

Handling only the case BP < AP < AB and missing the sliver near D where AB < AP < BP, which leaves pi/6 - sqrt3/8 and cannot even be written in the required form. (2025 AMC 10A #25)
Treating the three pairwise conditions as independent, or extrapolating the two-variable answer ((n-1)/n)^2, which gives a formula with n-1 instead of n-2. (2012 AMC 10A #25)
Thinking the third step is skipped or truncated when time runs out (it is completed), or coupling the step lengths to the times. (2022 AMC 10B #23)
Treating the corner triangles like the diamond, adding quarter-circles or strips along their legs, which lie outside the region the center can occupy. (2021 AMC 10B #23)

Problems, easiest first

2017 AMC 10A · #15Geometric Probability

Chloé chooses a real number uniformly at random from the interval $[0, 2017]$ . Independently, Laurent chooses a real number uniformly at random from the interval $[0, 4034]$ . What is the probability that Laurent's number is greater than Chloé's number?

2011 AMC 10B · #13Geometric Probability

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?

2003 AMC 10A · #12Geometric Probability

A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$ , $(4,0)$ , $(4,1)$ , and $(0,1)$ . What is the probability that $x<y$ ?

2024 AMC 10B · #14Geometric Probability

A dartboard is the region $B$ in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$ . A target $T$ is the region where $(x^2 + y^2 - 25)^2 \le 49$ . A dart is thrown and lands at a random point in B. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi$ , …

2023 AMC 10B · #19Geometric Probability

Sonya the frog chooses a point uniformly at random lying within the square $[0, 6]$ $\times$ $[0, 6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at random from {north, south, east, west}. All her choices are …

2020 AMC 10A · #16Geometric Probability

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$ . The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$ . (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to …

2014 AMC 10B · #19Geometric Probability

Two concentric circles have radii $1$ and $2$ . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

2011 AMC 10B · #16Geometric Probability

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?

2022 AMC 10B · #23Geometric Probability

Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3,$ Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1).$ During the $n$ th step of the process, Amelia moves $x_n$ units in the positive direction, using up …

2021 AMC 10B · #23Geometric Probability

A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square …

2018 AMC 10B · #22Geometric Probability

Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$ . Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?

2015 AMC 10A · #25Geometric Probability

Let $S$ be a square of side length $1$ . Two points are chosen at random on the sides of $S$ . The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$ , where $a$ , $b$ , and $c$ are positive integers with $\gcd(a,b,c)=1$ . What is $a+b+c$ ?

2011 AMC 10A · #20Geometric Probability

Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length $r$ is drawn in a clockwise direction. What is the probability that the two chords intersect?

2009 AMC 10B · #23Geometric Probability

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer …

2025 AMC 10A · #25Geometric Probability

A point $P$ is chosen at random inside square $ABCD$ . The probability that $\overline{AP}$ is neither the shortest nor the longest side of $\triangle APB$ can be written as $\frac{a + b \pi - c \sqrt{d}}{e}$ , where $a, b, c, d,$ and $e$ are positive integers, $\text{gcd}(a, b, c, e) = 1$ , and $d$ is not divisible by …

2012 AMC 10A · #25Geometric Probability

Real numbers $x$ , $y$ , and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$ . The probability that no two of $x$ , $y$ , and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$ . What is the smallest possible value of $n$ ?