All textbooks/Collection of Problems and Exercises in Probability Theory

Part I

Probability Space

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1 · Operations on Events

2 · The Classical Probability Space

3 · Geometric Probabilities

Section 1Operations on Events[25/64%]

Problem 1.1

(a)(b)(c)(d)

Among the students gathered for a lecture on probability theory, one is chosen at random. Let event $A$ be that the chosen student turns out to be male; event $B$ that he does not smoke; and event $C$ that he lives in a dormitory. Describe the event $AB\overline{C}$ . When do the following hold:

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(a)

the equality $AB\overline{C} = A$ ;

(b)

the inclusion $\overline{C} \subseteq B$ ;

(c)

the equality $\overline{A} = B$ ;

(d)

the equality $\overline{B} = B$ ?

Problem 1.2

A coin is tossed three times in a row. Construct the sample space $\Omega$ . Describe the event $A$ that heads came up at least twice.

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Problem 1.3

(a)(b)

Let $A$ , $B$ , and $C$ be events. What is the meaning of the equalities:

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(a)

$ABC = A$ ;

(b)

$A + B + C = A$ ?

Problem 1.4

(a)(b)(c)

Let $A$ , $B$ , and $C$ be events. Simplify the expressions:

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(a)

$(A+B)(B+C)$ ;

(b)

$(A+\overline{B})(\overline{A}+B)$ ;

(c)

$(A+B)(A+\overline{B})(\overline{A}+B)$ .

Problem 1.5

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)

Which of the following relations are correct:

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(a)

$(A+B) \setminus C = A + (B \setminus C)$ ;

(b)

$(A+B)\overline{C} = \overline{A}\, \overline{B}\, \overline{C}$ ;

(c)

$AB + C + CA \supseteq ABC$ ;

(d)

$(A+B) \setminus A = B$ ;

(e)

$A+B+C = ABC$ ;

(f)

$(A+B)\overline{C} = C \setminus (C(A+B))$ ;

(g)

$ABC = AB(B+C)$ ;

(h)

$A+B = (A \setminus AB) + B$ ;

(i)

$\overline{(A+B)C} = \overline{A}\, \overline{C} + \overline{B}\, \overline{C}$ ;

(j)

$ABC \subseteq A+B$ ;

(k)

$(AB+BC+CA) \subseteq (A+B+C)$ ?

Problem 1.6

(a)(b)(c)(d)(e)(f)(g)(h)(i)

Let $A$ , $B$ , and $C$ be arbitrary events. Find expressions for the events that, among $A$ , $B$ , and $C$ :

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(a)

only $A$ occurred;

(b)

$A$ and $B$ occurred, but not $C$ ;

(c)

all three events occurred;

(d)

at least one of these events occurred;

(e)

at least two events occurred;

(f)

exactly one of these events occurred;

(g)

exactly two events occurred;

(h)

none of the events occurred;

(i)

at most two events occurred.

Problem 1.7

(a)(b)(c)(d)(e)(f)

A worker has manufactured $n$ parts. Let event $A_i$ be that the $i$ -th manufactured part is defective. Write the event that:

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(a)

none of the parts is defective;

(b)

at least one part is defective;

(c)

exactly one part is defective;

(d)

at most two parts are defective;

(e)

at least two parts are not defective;

(f)

exactly two parts are defective.

Problem 1.8

The union $A \cup B$ of two events can be expressed as the union of two disjoint events, e.g., $A \cup B = A \cup (B\overline{A})$ . Express the union of three events $A$ , $B$ , and $C$ in a similar way.

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Problem 1.9

Prove that for any collection of events $A_1, \ldots , A_n$ the equalities

\overline{\bigcap _{i=1}^{n} A_i} = \bigcup _{i=1}^{n} \overline{A_i}, \qquad \overline{\bigcup _{i=1}^{n} A_i} = \bigcap _{i=1}^{n} \overline{A_i}

hold.

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Problem 1.10

(a)(b)(c)

Let $B$ and $C$ be two events. Set $A_n = B$ if $n$ is even and $A_n = C$ if $n$ is odd. Find the event that:

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(a)

infinitely many of the events $A_n$ occurred;

(b)

only finitely many of the events $A_n$ occurred;

(c)

all the events $A_n$ occurred, except possibly finitely many of them.

Problem 1.11

(a)(b)

Let $\Omega$ be a sample space, and let $A$ be an arbitrary subset of it with $A \neq \Omega$ and $A \neq \emptyset$ . Prove that the following collections of subsets of $\Omega$ are $\sigma$ -algebras:

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(a)

$\mathcal{F} = \left\{ \Omega , \emptyset \right\}$ (this $\sigma$ -algebra is called trivial);

(b)

$\mathcal{F} = \left\{ \Omega , \emptyset , A, \overline{A}\right\}$ .

Problem 1.12

Prove that if the sample space $\Omega$ consists of $n$ elements, then the set of all subsets of $\Omega$ contains exactly $2^n$ elements.

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Problem 1.13

(a)(b)

A coin is tossed three times in a row. Are the following systems of subsets $\sigma$ -algebras:

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(a)

$\emptyset$ , $\Omega$ , $\left\{ \text{HHH, HHT, HTH, HTT}\right\}$ , $\left\{ \text{THH, THT, TTH, TTT}\right\}$ ;

(b)

$\emptyset$ , $\Omega$ , $\left\{ \text{HHH, HTH, THT, TTT}\right\}$ , $\left\{ \text{THH, THT, TTH, TTT}\right\}$ ?

Problem 1.14

Let $\Omega = \mathbb {R}$ . Prove that the collection of subsets $\mathcal{A} = \left\{ \Omega , \emptyset , [0,1], \left\{ 0\right\} \right\}$ is not a $\sigma$ -algebra. Construct the minimal $\sigma$ -algebra containing $\mathcal{A}$ .

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Problem 1.15

(a)(b)(c)(d)(e)(f)

Let $\mathcal{F}$ be the collection of all subsets $A \subseteq \mathbb {R}$ such that either $A$ or $\mathbb {R} \setminus A$ contains finitely many elements. Is $\mathcal{F}$ an algebra of sets? Find the smallest $\sigma$ -algebra generated by $\mathcal{F}$ . Are the following functions measurable with respect to $\sigma (\mathcal{F})$ :

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(a)

$f(x) = x$ ;

(b)

$f(x) \equiv 17$ ;

(c)

$f(x) = \; \mathbb {1}_{\mathbb {Z}}(x)$ ;

(d)

$f(x) = \; \mathbb {1}_{[0,1]}(x)$ ;

(e)

$f(x) = \; \mathbb {1}_{\left\{ 0\right\} }(x)$ ;

(f)

$f(x) = x \; \mathbb {1}_{\mathbb {Z}}(x)$ ?

Problem 1.16

(a)(b)(c)

Let $\mathcal{F}$ be the collection of intervals in $\mathbb {R}$ of the form $[0, 2^{-n})$ , where $n \in \mathbb {Z}_{+}$ .

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(a)

Is $\mathcal{F}$ a $\sigma$ -algebra of subsets?

(b)

Find the smallest $\sigma$ -algebra generated by $\mathcal{F}$ .

(c)

Which functions from $\mathbb {R}$ to $\mathbb {R}$ are measurable with respect to $\sigma (\mathcal{F})$ ?

Problem 1.17

(a)(b)(c)

Let $\mathcal{F}$ be the collection of intervals in $\mathbb {R}$ of the form $[0, n)$ , where $n \in \mathbb {Z}_{+}$ .

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(a)

Is $\mathcal{F}$ a $\sigma$ -algebra of subsets?

(b)

Find the smallest $\sigma$ -algebra generated by $\mathcal{F}$ .

(c)

Which functions from $\mathbb {R}$ to $\mathbb {R}$ are measurable with respect to $\sigma (\mathcal{F})$ ?

Problem 1.18

(a)(b)(c)(d)

Let $\mathcal{F}_1$ be the collection of all subsets of $\mathbb {R}^{2}$ of the form $B \times \mathbb {R}$ , where $B \subseteq \mathbb {R}$ is Borel-measurable, and let $\mathcal{F}_2$ be the collection of those of the form $\mathbb {R} \times B$ .

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(a)

Are $\mathcal{F}_1$ and $\mathcal{F}_2$ $\sigma$ -algebras of subsets?

(b)

Which events of $\mathcal{F}_2$ are measurable with respect to $\mathcal{F}_1$ ?

(c)

Which functions from $\mathbb {R}^{2}$ to $\mathbb {R}$ are measurable with respect to $\mathcal{F}_1$ ?

(d)

Find the smallest $\sigma$ -algebra generated by $\mathcal{F}_1 \cup \mathcal{F}_2$ .

Problem 1.19

(a)(b)(c)(d)(e)(f)

For any events $A$ and $B$ , prove that:

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(a)

$\mathbb {P}\left(\overline{A}\right) = 1 - \mathbb {P}\left(A\right)$ ;

(b)

if $A \subseteq B$ , then $\mathbb {P}\left(B \setminus A\right) = \mathbb {P}\left(B\right) - \mathbb {P}\left(A\right)$ ;

(c)

if $A \subseteq B$ , then $\mathbb {P}\left(A\right) \leqslant \mathbb {P}\left(B\right)$ ;

(d)

$\mathbb {P}\left(A \cup B\right) = \mathbb {P}\left(A\right) + \mathbb {P}\left(B\right) - \mathbb {P}\left(AB\right)$ ;

(e)

$1 - \mathbb {P}\left(A\right) - \mathbb {P}\left(B\right) \leqslant \mathbb {P}\left(AB\right) \leqslant 1$ ;

(f)

$\mathbb {P}\left(\overline{A}\, \overline{B}\right) = 1 - \mathbb {P}\left(A\right) - \mathbb {P}\left(B\right) + \mathbb {P}\left(AB\right)$ .

Problem 1.20

(a)(b)

For any collection of events $A_1, \ldots , A_n$ , prove the following equalities (the inclusion-exclusion formulas):

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(a)

\mathbb {P}\left(\bigcup _{i=1}^{n} A_i\right) = \sum _{i=1}^{n} \mathbb {P}\left(A_i\right) - \sum _{1 \leqslant i < j \leqslant n} \mathbb {P}\left(A_i A_j\right) + \sum _{1 \leqslant i < j < k \leqslant n} \mathbb {P}\left(A_i A_j A_k\right) - \ldots + (-1)^{n+1} \mathbb {P}\left(\bigcap _{i=1}^{n} A_i\right);

(b)

\mathbb {P}\left(\bigcap _{i=1}^{n} A_i\right) = \sum _{i=1}^{n} \mathbb {P}\left(A_i\right) - \sum _{1 \leqslant i < j \leqslant n} \mathbb {P}\left(A_i \cup A_j\right) + \sum _{1 \leqslant i < j < k \leqslant n} \mathbb {P}\left(A_i \cup A_j \cup A_k\right) - \ldots + (-1)^{n+1} \mathbb {P}\left(\bigcup _{i=1}^{n} A_i\right).

Problem 1.21

Let $\mathbb {P}\left(A_n\right) = 0$ for all $n$ . Prove the equality

\mathbb {P}\left(\bigcup _{n=1}^{\infty } A_n\right) = 0.

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Problem 1.22

Let $\mathbb {P}\left(A_n\right) = 1$ for all $n$ . Prove the equality

\mathbb {P}\left(\bigcap _{n=1}^{\infty } A_n\right) = 1.

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Problem 1.23

Prove that for any sequence of events $\left\{ A_n\right\}$ the inequality

\mathbb {P}\left(\bigcup _{i=1}^{\infty } A_i\right) \leqslant \sum _{i=1}^{\infty } \mathbb {P}\left(A_i\right)

holds.

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Problem 1.24

Consider a sequence of events $\left\{ A_n\right\}$ and denote by $A$ the event that infinitely many of the events $A_n$ occur, i.e.,

A = \left\{ \omega : \sum _{n=1}^{\infty } \; \mathbb {1}_{A_n}(\omega ) = \infty \right\} .

Prove that

A = \bigcap _{n=1}^{\infty } \bigcup _{k=n}^{\infty } A_n.

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Problem 1.25

In the setting of the previous problem, prove that if the series $\sum_{n=1}^{\infty } \mathbb {P}\left(A_n\right)$ converges, then $\mathbb {P}\left(A\right) = 0$ (the Borel--Cantelli lemma).

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Section 2The Classical Probability Space[51/100%]

Problem 2.1

On a shelf, $n$ books are arranged in random order, among them a two-volume set by D. London. Assume that all arrangements of the books are equally likely. Find the probability that both volumes end up next to each other, with the first volume to the left of the second.

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Problem 2.2

(a)(b)

$N$ people sit down at random around a round table. Find the probability that:

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(a)

friends $A$ and $B$ sit next to each other, with $B$ to the left of $A$ ;

(b)

friends $A$ , $B$ , and $C$ sit next to each other, with $A$ to the right of $B$ and $C$ to the left.

Problem 2.3

$N$ people, among whom are $A$ and $B$ , are placed at random in a row. What is the probability that there are exactly $r$ people between $A$ and $B$ ?

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Problem 2.4

(a)(b)

The numbers $1, \ldots , n$ are arranged in random order. Find the probability that the numbers

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(a)

1 and 2;

(b)

1, 2, and 3

end up next to each other in the stated order.

Problem 2.5

A child is playing with 10 letter tiles: А, А, А, Е, И, К, М, М, Т, Т (the Cyrillic letters spelling МАТЕМАТИКА, i.e., ``mathematics''). What is the probability that, arranging the letters in a row at random, the child gets the word МАТЕМАТИКА?

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Problem 2.6

(a)(b)(c)(d)

Two dice are thrown. Let event $A$ be that the sum of the points shown is odd, and event $B$ that at least one of the dice shows a one. Describe the events:

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(a)

$A \cap B$ ;

(b)

$A \cap \overline{B}$ ;

(c)

$A \cup B$ ;

(d)

$\overline{A} \cap B$ .

Find their probabilities.

Problem 2.7

Find the probability of rolling 12 points at least once in $n$ throws of two dice.

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Problem 2.8

(a)(b)(c)

Three coins are tossed. Find the probabilities of the events:

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(a)

$A = \left\{ \text{the first coin came up heads}\right\}$ ;

(b)

$B = \left\{ \text{exactly two heads came up}\right\}$ ;

(c)

$C = \left\{ \text{at most two heads came up}\right\}$ .

Problem 2.9

A cube with all its faces painted is sawn into 1000 unit cubes of the same size. Determine the probability that a cube drawn at random has exactly two painted faces.

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Problem 2.10

Four dice are thrown. Find the probability that they all show the same number of points.

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Problem 2.11

(a)(b)

Five dice are thrown. Find the probability that at least

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(a)

two;

(b)

three

of them show matching faces.

Problem 2.12

(a)(b)(c)

Five people enter the elevator of a 9-story building on the first floor. Each of them is equally likely to get off at any floor from the second up. Find the probability that all five get off:

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(a)

on the fifth floor;

(b)

simultaneously (on the same floor);

(c)

on different floors.

Problem 2.13

(a)(b)(c)(d)(e)(f)(g)

What is the probability that in the four-digit license plate number of a car chosen at random in a large city:

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(a)

all the digits are different;

(b)

there are two pairs of matching digits of the form $AABB$ , including numbers of the form $AAAA$ ;

(c)

there are two pairs of matching digits of the form $AABB$ , excluding numbers of the form $AAAA$ ;

(d)

exactly two digits match;

(e)

exactly three digits match;

(f)

all the digits are the same;

(g)

the sum of the first two digits equals the sum of the last two?

Remark.

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The answers are given under the assumption that the number 0000 is possible.

Problem 2.14

(a)(b)(c)(d)(e)

Find the probability that among $r$ randomly chosen (with replacement) digits:

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(a)

0 does not occur;

(b)

1 does not occur;

(c)

neither 0 nor 1 occurs;

(d)

there are no two equal digits;

(e)

at least one of the two digits 0 or 1 does not occur.

Problem 2.15

The digits 1, 2, 3, 4, and 5 are written on five cards. Three cards are drawn at random, one at a time, and placed side by side from left to right. What is the probability that the resulting number is even?

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Problem 2.16

(a)(b)(c)

From the digits 1, 2, 3, 4, and 5, one is chosen first, and then a second is chosen from the remaining four. Find the probability that:

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(a)

the first;

(b)

the second;

(c)

both

time(s) an odd digit is chosen.

Problem 2.17

(a)(b)(c)(d)(e)(f)(g)(h)

A deck of playing cards (52 cards, 4 suits of 13 cards each) is thoroughly shuffled. Six cards are drawn at random (without replacement). Describe the sample space, and also find the probability that among these cards:

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(a)

the king of spades is among them;

(b)

cards of all four suits are among them;

(c)

there are exactly 5 cards of one suit;

(d)

the king of spades is the first card drawn;

(e)

the king of spades is the fifth card drawn;

(f)

there are exactly 5 face cards and 2 red cards;

(g)

there are at least 2 diamond cards;

(h)

there are at least 2 red cards.

Problem 2.18

(a)(b)(c)(d)(e)(f)(g)(h)

Solve problem 2.17 for the case where 7 cards are drawn from the deck: a deck of playing cards (52 cards, 4 suits of 13 cards each) is thoroughly shuffled; describe the sample space and find the probability that among these cards:

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(a)

the king of spades is among them;

(b)

cards of all four suits are among them;

(c)

there are exactly 5 cards of one suit;

(d)

the king of spades is the first card drawn;

(e)

the king of spades is the fifth card drawn;

(f)

there are exactly 5 face cards and 2 red cards;

(g)

there are at least 2 diamond cards;

(h)

there are at least 2 red cards.

Problem 2.19

(a)(b)(c)

Three cards are drawn at random from a deck of 52. Find the probability that:

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(a)

exactly one ace is among them;

(b)

at least one ace is among them;

(c)

they are a three, a seven, and an ace (in any order).

Problem 2.20

(a)(b)(c)

A deck of cards (36 cards) is split at random into two equal stacks. What is the probability that:

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(a)

each stack ends up with exactly two aces;

(b)

one stack ends up with all four aces;

(c)

the stacks end up with an equal number of red cards?

Problem 2.21

What is the smallest number of cards that must be drawn from a deck (52 cards) so that, with probability greater than $1/2$ , at least two cards of the same rank appear among them?

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Problem 2.22

A box contains 10 red and 6 blue buttons. What is the probability that two randomly drawn buttons are the same color?

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Problem 2.23

A lottery distributes $n$ tickets, of which $m$ are winning. Someone buys $k$ tickets. What is the probability that at least one of them is winning?

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Problem 2.24

(a)(b)

A batch of $N$ items contains $M$ defective ones. What is the probability that among $n$ ( $n < N$ ) items chosen at random from this batch, there are:

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(a)

exactly $m$ defective ones ( $m < M$ );

(b)

no more than $m$ defective ones?

Problem 2.25

(a)(b)(c)(d)

A participant in the ``Sportloto'' lottery names six out of 49 sports. The prize is determined by how many names they guessed correctly out of the six other names, which are determined at the moment of the drawing by a special device implementing a random choice. What is the probability that the participant guesses:

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(a)

all six names;

(b)

five names;

(c)

two names;

(d)

none of the names?

Problem 2.26

A box contains 90 good and 10 defective screws. What is the probability that among ten screws drawn at random, none is defective?

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Problem 2.27

In a parking lot, $n$ spaces are arranged in a single row. A car arriving at the lot takes one of the free spaces (not at either end). When the owner returns, they find that exactly $r$ spaces are still occupied. What is the probability that both neighboring spaces are free?

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Problem 2.28

(a)(b)(c)(d)

A closet contains $n$ pairs of shoes. $r$ shoes are chosen at random ( $r \leqslant n$ ). What is the probability that among them:

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(a)

there is not a single complete pair;

(b)

there is exactly one complete pair;

(c)

there are exactly two complete pairs;

(d)

there is at least one complete pair?

Problem 2.29

A group of $2n$ girls and $2n$ boys is split into two equal subgroups. Find the probability that each subgroup contains the same number of boys and girls.

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Problem 2.30

A group of 3 girls and $3n$ boys is split into three equal subgroups. What is the probability that all the girls end up in different subgroups?

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Problem 2.31

(a)(b)

A sleeping car (9 compartments with 4 berths each) has seven tickets sold to seven passengers. Find the probability that exactly the following end up occupied:

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(a)

two compartments;

(b)

three compartments.

Problem 2.32

From the set $\left\{ 1, 2, \ldots , n\right\}$ , two numbers are drawn one after another without replacement. What is the probability that the difference between the first number drawn and the second is at least $m$ , $0 < m < n$ ?

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Problem 2.33

(a)(b)

Find the probability that:

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(a)

the birthdays of 12 people fall in 12 different months of the year (assume all months equally likely);

(b)

the birthdays of 6 people fall in exactly two months.

Problem 2.34

$n$ strangers gather together. Find the probability that at least two of them share the same birthday.

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Problem 2.35

Ten manuscripts are distributed among 30 folders (three folders per manuscript). Find the probability that among 6 randomly chosen folders, no manuscript is contained in its entirety.

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Problem 2.36

Someone writes letters to $n$ addressees, places one letter in each envelope, and then writes one of the $n$ addresses at random on each envelope. Find the probability that at least one letter reaches its intended addressee.

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Problem 2.37

(a)(b)

An urn contains $n$ tickets numbered from 1 to $n$ . The tickets are drawn at random one at a time (without replacement) until the urn is empty. What is the probability that:

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(a)

at least once the number of the ticket drawn matches the number of the trial;

(b)

the numbers of the first $m$ tickets drawn appear in increasing order?

Problem 2.38

A schoolboy, wanting to play a joke on his classmates, gathered all the coats in the cloakroom and then hung them back up in random order. What is the probability $p_n$ that at least one coat ends up on its original hook, if the cloakroom has $n$ hooks and $n$ coats were hanging on them? Find the limit of $p_n$ as $n \to \infty$ .

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Problem 2.39

(a)(b)

A group of $n$ people decides to exchange gifts, so each person brings a gift. All the gifts are put together, mixed up, and distributed at random among the participants. Find the probability that:

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(a)

at least one gift returns to its own owner;

(b)

a particular person receives their own gift.

Compare the behavior of these probabilities as $n$ increases.

Problem 2.40

Two dice are thrown $r$ times, $r \geqslant 6$ . Find the probability that each of the six combinations $(1,1), \ldots , (6,6)$ appears at least once.

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Problem 2.41

(a)(b)

Each of $n$ sticks breaks into two pieces --- a long one and a short one. Then the $2n$ pieces are randomly joined into $n$ pairs, each pair forming a new ``stick''. Find the probability that:

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(a)

the pieces are joined in their original order;

(b)

all the long pieces end up joined with short ones.

Problem 2.42

(a)(b)

Suppose there are $n$ cells into which $r$ numbered particles are placed at random. What is the probability that:

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(a)

exactly $s$ particles land in the $k$ -th cell;

(b)

at least one cell remains empty?

Problem 2.43

From the set of numbers $1, 2, \ldots , N$ , $n$ numbers are randomly drawn without replacement, $n \leqslant N$ , and arranged in increasing order: $x_1 < x_2 < \ldots < x_n$ . What is the probability that $x_m < M$ ? Find the limit of this probability as $M, N \to \infty$ so that $M/N \to \alpha > 0$ .

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Problem 2.44

(a)(b)

What is the probability that two throws of three dice give the same result, if:

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(a)

the dice are distinguishable;

(b)

the dice are indistinguishable?

Problem 2.45

(a)(b)

Find the probability that when $n$ distinct balls are placed at random into $n$ boxes,

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(a)

no box remains empty;

(b)

exactly one box remains empty.

Problem 2.46

Suppose there are $n$ cells into which $r$ indistinguishable particles are placed at random. This scheme corresponds to photons, atomic nuclei, and atoms containing an even number of elementary particles. They obey Bose--Einstein statistics, in which only distinguishable arrangements of particles among cells are considered, and each is assigned equal probability. Find this probability. Find the probability that the first cell contains exactly $k$ particles.

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Problem 2.47

Suppose there are $n$ cells and $r$ indistinguishable particles. The particles are placed into the cells so that no cell can hold more than one particle. This scheme corresponds to electrons, protons, and neutrons. They obey Fermi--Dirac statistics, in which each such arrangement of particles among cells is assigned equal probability. Find this probability.

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Problem 2.48

(a)(b)

A stream of $k$ particles is captured by a system of $n$ particle-detecting counters. Each particle is equally likely to land in any of the counters.

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(a)

What is the probability $p(k_1, \ldots , k_n)$ that the first counter registers $k_1$ particles, the second $k_2$ particles, and so on, where $k_1 + \ldots + k_n = k$ ?

(b)

Let $k/n = l$ be an integer. For what values of $k_1, \ldots , k_n$ does the probability $p(k_1, \ldots , k_n)$ attain its maximum?

Problem 2.49

(a)(b)

A city has $n+1$ residents. One of them, having learned some news, tells it to another, that person tells a third, and so on, with the news being passed on as follows: the person who has been told the news chooses one of the $n$ residents at random and tells it to them, that person does the same, and so on. Find the probability that the news is passed on $r$ times:

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(a)

without returning to the person who first learned it;

(b)

without being told again to anyone who already knows it.

Problem 2.50

From the set $\left\{ 1, 2, \ldots , n\right\}$ , a number $\xi$ is chosen at random. Find the limit as $n \to \infty$ of the probability that $\xi^2 - 1$ is divisible by 10.

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Problem 2.51

(a)(b)

From the set $\left\{ 1, 2, \ldots , n\right\}$ , $n \geqslant 4$ , two numbers $\xi$ and $\eta$ are chosen at random (with replacement). Which probability is larger:

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(a)

$\mathbb {P}\left(\xi^2 - \eta^2 \text{ is divisible by } 2\right)$ or $\mathbb {P}\left(\xi^2 - \eta^2 \text{ is divisible by } 3\right)$ ;

(b)

$\mathbb {P}\left(\xi^3 - \eta^3 \text{ is divisible by } 2\right)$ or $\mathbb {P}\left(\xi^3 - \eta^3 \text{ is divisible by } 3\right)$ ?

Section 3Geometric Probabilities[20/100%]

Problem 3.1

(a)(b)(c)(d)(e)(f)

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

(a)(b)(c)(d)(e)

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 $\xi^2 + \eta^2 \leqslant n^2$ .