Combinatorics · Chapter 8

Geometric Probability

When outcomes form a continuum, count them by measuring: lengths, areas and volumes turn waiting times, random points and dropped coins into probabilities.

What's this chapter about?

8.0 Probability as Area
8.1 Geometric Probability in 3D
8.2 Practice Problems

You'll need

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8.00%

Probability as Area

In this section you will be able to turn a probability with infinitely many outcomes into a ratio of areas.

Alex and Bob each show up at a random moment between 3 and 4 PM, and each waits 15 minutes. There are infinitely many ways this can go. So how can you count them?

You can’t list the outcomes, but you can draw them. Two random times make a point in a square. The outcomes you want make a region. The probability is the region’s share of the square’s area.

Look: the hard part is never the arithmetic. It is finding the region. Try the extreme cases first, mark them on the picture, then fill in the shape between them.

Exploreplay first, formulas later

Try this: Drag w from 15 up to 30 minutes. How does the band grow? Predict the probability at w = 30 before you look.
Try this: Throw 1000 darts, then 10,000. How close does the estimate get to the exact value?
Try this: Pick Obtuse (Example 8.3). Which part of the square makes a triangle at all, and which part of that is obtuse?
Try this: Pick Coin drop. Why does the dashed box shrink when the coin gets bigger?

Name it

Definition 8.0.1 · Geometric Probability

Geometric probability is a way to calculate probability by measuring the number of outcomes geometrically, in terms of length, area, or volume. The key to solving geometric probability is

Try a few examples for the different cases, and always mark the extreme cases.
Figure out the region the shape maps out.
Use geometry to find the area of this region.

P = \frac{\text{size of the good region}}{\text{size of the whole sample space}}

Remark 8.0.2

Geometric probability can be useful when the number of possible outcomes is infinite.

You can't list infinitely many outcomes, but you can measure them. Throwing random darts at the picture shows the area ratio really is the probability: the share of darts in the shaded region settles down near it.

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Example 8.1

Alex and Bob arrive at a random time from 3 PM to 4 PM. They will each wait for 15 minutes before leaving. What is the probability that they will meet?

1
What must happen for the 2 people to meet?

Predict first:
Click a choice or type its letter (A–E).

Example 8.2

AMC 10

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? (Assume they cannot be equal)

1
Two random numbers, so draw a picture. What shape is the sample space if Chloé's number is $x$ and Laurent's is $y$ ?

Predict first:
Click a choice or type its letter (A–E).

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Example 8.3

AMC 10

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?

(A) $0.21$
(B) $0.25$
(C) $0.29$
(D) $0.50$
(E) $0.79$

1
Which of $x$ , $y$ , $1$ is the longest side?

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Example 8.4

AMC 10

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 and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}\left(a + b\sqrt{2} + \pi\right)$ , where $a$ and $b$ are positive integers. What is $a + b$ ?

1
Track the coin's center. The coin (radius $\frac12$ ) must stay inside the square. Where can the center land, and what is that area?

Predict first:
Your prediction

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Example 8.5

AMC 10/12

Let $S$ be a square of side length $1$ . Two points are chosen independently at random on the sides of $S$ . The probability that the straight-line distance between the points is at least $\frac12$ is $\frac{a - b\pi}{c}$ , where $a$ , $b$ , and $c$ are positive integers with $\gcd(a, b, c) = 1$ . What is $a + b + c$ ?

1
Break it into cases by where the second point lands. Case 1: both points on the same side. How likely is this case?

Predict first:
Your prediction

Try it

Problem 8.2.1

This problem also appears in the book as 8.1.

Alex and Bob decide to meet at the park between 3 and 4 pm. They both will arrive at a random time during the hour but will only wait up to 15 minutes for the other person and then leave the park. What is the probability that they will meet?

Contest cornerreal contest problems

Problem 8.2.3

AMC 10

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 the choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?

Problem 8.2.4

AMC 10

Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0, 1]$ . Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x - y| > \frac12$ ?

Problem 8.2.6

AMC 10A

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, $\gcd(a, b, c, e) = 1$ , and $d$ is not divisible by the square of a prime. What is $a + b + c + d + e$ ?

Problem 8.2.7(additional)

AIME

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40\%$ , and $m = a - b\sqrt{c}$ , where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c$ .

Problem 8.2.8(additional)

AIME

A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$ .

Checkpoint for section 8.0

Get 2 of 3 right to master this section. One try each, no hints.

8.10%

Geometric Probability in 3D

In this section you will be able to handle three random numbers by slicing the unit cube and adding up the slices.

Ant Amelia picks three random step lengths. Will they add up to more than 1? Three random numbers make a point in a cube, so now the probability is a volume.

Volumes are harder to see than areas. The trick is to slice. Fix the last variable at height $t$ and the slice is a 2-D region you already know how to measure. Stack the slices from $t = 0$ to $t = 1$ and you have the volume.

Look: the corner where $x + y + z \le 1$ is a pyramid with volume $\frac13 \cdot \frac12 \cdot 1$ . That one number answers a surprising number of contest problems.

Exploreplay first, formulas later

Try this: Slide t from 0 to 1. When does the cross-section vanish?
Try this: Why is the corner x + y + z ≤ 1 exactly 1/6 of the cube? Compare it with the shaded area under the graph.
Try this: Switch to Largest > 2 × others. Why does the slice change shape at t = 1/2?

Watch itpredict, then reveal

Example 8.6

AMC 10/12

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 $t_n$ minutes. If the total elapsed time has exceeded $1$ minute during the $n$ th step, she stops at the end of that step; otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia's position when she stops will be greater than $1$ ?

1
Can Amelia stop after just one step?

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Contest cornerreal contest problems

Problem 8.2.5

AMC 12

Three real numbers are chosen independently and uniformly at random between 0 and 1. What is the probability that the greatest of these three numbers is greater than 2 times each of the other two numbers? (In other words, if the chosen numbers are $a \ge b \ge c$ , then $a > 2b$ .)

Checkpoint for section 8.1

Get 2 of 3 right to master this section. One try each, no hints.

8.2

Practice Problems

In this section you will be able to practice a mixed problem from this chapter's problem set.

Not every probability in this chapter is geometric. This octagon problem counts triangles instead of measuring areas, so pick the tool that fits.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 8.2.2

AMC

This problem also appears in the book as 7.6.2.

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?