Math Problems Vault

Problem 1.2.1

A group has 25 students. Assume that each student's birthday is random (let there be 365 days in a year). Find the probability that at least two people share a birthday.

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Problem 1.2.2

Some residents of Dolgoprudny consider a tram ticket "lucky" if the sum of the first three digits of its six-digit number matches the sum of the last three digits. Find the probability of getting a "lucky" ticket.

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Problem 1.2.3

(a)(b)

On an $n \times n$ chessboard, $n$ rooks are placed at random. Find the probabilities of the following events:

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(a)

$A=\left\{ \text{ the rooks do not attack each other }\right\}$ ;

(b)

$B=\left\{ \text{ the rooks do not attack each other and there are no pieces on the main diagonal }\right\}$ .

Problem 1.2.4

(a)(b)(c)

A random point $A$ has a uniform distribution in a rectangle with sides 1 and 2. Find the probabilities of the following events:

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(a)

the distance from the point $A$ to the nearest diagonal of the rectangle does not exceed $x$ ;

(b)

the distance from the point $A$ to any side of the rectangle does not exceed $x$ ;

(c)

the distance from the point $A$ to the nearest side of the rectangle is less than the distance from $A$ to the nearest diagonal.

Problem 1.2.5

(a)(b)(c)

A chord is drawn at random in a circle of radius $R$ . Denote its length by $\xi$ . Find the probability $\mathbb {P}\left(\xi >\sqrt{3} R\right)$ , if

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(a)

the midpoint of the chord is uniformly distributed in the circle;

(b)

the direction of the chord is fixed, and its midpoint is uniformly distributed on the diameter perpendicular to its direction;

(c)

one end of the chord is fixed, and the other is uniformly distributed on the circle.

Problem 1.2.6

Find the probability that a triangle can be formed from three segments of length at most 1 chosen at random.

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Problem 1.2.7

In the card game poker, a player receives 5 cards from a deck of 52 cards. The player's task is to assemble the strongest possible combination of cards. The combinations are as follows:

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(a)

a pair - two cards of the same rank;

(b)

two pairs - two cards of one rank, two cards of another;

(c)

three of a kind - three cards of the same rank;

(d)

a straight - five cards with consecutive ranks (it is assumed that a two follows an ace in rank);

(e)

a flush - all cards of the same suit;

(f)

full house - three cards of one rank, two cards of another;

(g)

four of a kind - four cards of the same rank;

(h)

a straight flush - five cards with consecutive ranks of the same suit;

(i)

a royal flush - ace, king, queen, jack and ten of the same suit.

Find the probability of obtaining each of the listed combinations when the cards are dealt at random. Compute the probability that none of the above combinations occurs.

Problem 1.2.8

In web search, when solving various machine learning problems, a statistical method called bootstrapping is often used. It works as follows. Suppose we have $N$ web pages. We want to find out how robust our algorithm is. To do this, $N$ pages are chosen at random (some may coincide) a large number $M$ of times (in this case one says that $M$ samples of size $N$ are generated). If the selection is performed with order taken into account, find the probability that the first page occurs $k$ times in one such sample, and the second $-r$ times.

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Problem 1.2.9

$n+1$ points are chosen at random on an $n$ -dimensional sphere. Find the probability that their convex hull does not contain the center of the sphere.

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