Math Problems Vault

Problem 3.1

A point is thrown at random into the square with vertices $(0,0)$ , $(0,1)$ , $(1,0)$ , $(1,1)$ . Let $(\xi , \eta )$ be its coordinates. Prove that for any $0 \leqslant x, y \leqslant 1$ ,

\mathbb {P}\left(\xi < x, \eta < y\right) = \mathbb {P}\left(\xi < x\right)\mathbb {P}\left(\eta < y\right) = xy.

Find:

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(a)

$\mathbb {P}\left(\left|\xi - \eta \right| < z\right)$ ;

(b)

$\mathbb {P}\left(\min (\xi , \eta ) < z\right)$ ;

(c)

$\mathbb {P}\left((\xi + \eta )/2 < z\right)$ ;

(d)

$\mathbb {P}\left(\xi \eta < z\right)$ ;

(e)

$\mathbb {P}\left(\max (\xi , \eta ) < z\right)$ ;

(f)

$\mathbb {P}\left(\xi + 2\eta < z\right)$ .

Problem 3.2

(a)(b)(c)

A random point $A$ is chosen at random in a rectangle with sides 1 and 2. Find the probability that the distance from $A$ to:

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(a)

the nearest side of the rectangle does not exceed $x$ ;

(b)

every side of the rectangle does not exceed $x$ ;

(c)

every diagonal of the rectangle does not exceed $x$ .

Problem 3.3

Two people, $A$ and $B$ , agree to meet at a certain place between two and three o'clock in the afternoon. Whoever arrives first waits for the other for 10 minutes, after which they leave. What is the probability that $A$ and $B$ meet, if each of them may arrive at any time during the given hour?

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Problem 3.4

Two people, $A$ and $B$ , agree to meet at a certain place between two and three o'clock in the afternoon. Person $A$ waits for the other for 10 minutes, after which they leave; person $B$ waits for 15 minutes. What is the probability that $A$ and $B$ meet, if each of them may arrive at any time during the given hour?

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Problem 3.5

Parallel lines a distance $2a$ apart are drawn on a plane. A needle of length $2r < 2a$ is thrown onto the plane at random. What is the probability that the needle crosses one of the lines?

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Problem 3.6

(a)(b)

A needle of length $2r < 2a$ is thrown at random onto an infinite chessboard with cell side $2a$ . What is the probability that the needle:

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(a)

lands entirely inside one cell;

(b)

crosses one of the lines?

Problem 3.7

Parallel lines a distance $2a$ apart are drawn on a plane. A coin of radius $r < a$ is thrown onto the plane at random. What is the probability that the coin crosses one of the lines?

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Problem 3.8

(a)(b)

A coin of radius $r < a$ is thrown at random onto an infinite chessboard with cell side $2a$ . Find the probability that:

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(a)

the coin lands entirely inside one cell;

(b)

the coin crosses at most one side of one cell.

Problem 3.9

(a)(b)

A rod of length $l$ is broken at two randomly chosen points. What is the probability that the resulting segments can form:

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(a)

a triangle;

(b)

a right triangle?

Problem 3.10

Three points are thrown at random onto the segment $[0,a]$ . Find the probability that the segments equal to the distances from the point 0 to the points of impact can form a triangle.

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Problem 3.11

A rod of length $l$ is broken at random into two pieces, after which the larger piece is broken again in two at a randomly chosen point. Find the probability that the resulting pieces can form a triangle.

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Problem 3.12

Three points $A$ , $B$ , and $C$ lie on a single line, with point $B$ between points $A$ and $C$ , the length of segment $AB$ equal to $a$ , and the length of segment $BC$ equal to $b$ . On each of the segments $AB$ and $BC$ , a point is thrown at random. Find the probability that a triangle can be formed from the following three segments: from point $A$ to the first thrown point; between the two thrown points; from the second thrown point to point $C$ .

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Problem 3.13

Three points $A$ , $B$ , and $C$ are chosen at random on a circle. Find the probability that the triangle $ABC$ is:

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(a)

acute;

(b)

obtuse;

(c)

right;

(d)

equilateral;

(e)

isosceles.

Problem 3.14

(a)(b)(c)

Three points are thrown at random into a square. Find the probability that they form the vertices of:

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(a)

some triangle;

(b)

an equilateral triangle;

(c)

a right triangle.

Problem 3.15

$n$ points are scattered randomly and independently of each other inside a ball of radius $R$ . What is the probability that the distance from the center to the nearest point is at least $r$ ? Find the limit of this probability as $R, n \to \infty$ , if $n/R^3 \to c$ .

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Problem 3.16

(a)(b)(c)

A point $(\xi , \eta )$ is chosen at random in the square $[0,1]^2$ . What is the probability that the equation $x^2 + \xi x + \eta = 0$ has:

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(a)

real roots;

(b)

positive roots;

(c)

roots of opposite sign?

Problem 3.17

A point $(\xi , \eta , \zeta )$ is chosen at random in the cube $[0,1]^3$ . What is the probability that the roots of the equation $\xi x^2 + \eta x + \zeta = 0$ are real?

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Problem 3.18

How thick must a coin be so that the probability of it landing on its edge equals $1/3$ ?

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Remark.

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Do not take rotational moment of inertia into account; assume that the coin's angle of tilt while falling takes different values with equal probability.

Problem 3.19

(a)(b)(c)

A chord is chosen at random for a given circle. Find the probability that this chord is longer than a side of the equilateral triangle inscribed in the given circle, if:

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(a)

the chord is determined by its midpoint, which is chosen at random inside the given disk;

(b)

one endpoint of the chord is a fixed point on the circle, and the other endpoint is chosen at random on the circle;

(c)

the chord is determined by its midpoint, which is chosen at random on a diameter of the disk.

Give an example of at least one more method of choosing a random chord.

Problem 3.20

(a)(b)

From the set $\left\{ 1, 2, \ldots , n\right\}$ , two numbers $\xi$ and $\eta$ are chosen at random:

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(a)

with replacement;

(b)

without replacement.

Find the limit as $n \to \infty$ of the probability of the event $\xi^2 + \eta^2 \leqslant n^2$ .