Application of the Basic Theorems. Bayes' Formula

In recording the surnames of the members of a certain assembly, whose total number is 360, it turned out that the initial letter of the surname was $A$ for seven, $E$ for five, $И$ for eight, $O$ for nine, $У$ for four, $Ю$ for two, and for all the rest the surname began with a consonant. Determine the probability $\mathbb {P}\left(P\right)$ that the surname of a member of this assembly begins with a vowel.

From a box containing 20 white balls and 2 black ones, $m$ balls are drawn, with the ball being put back after each draw; determine the smallest value $x_{0}$ of the number $m$ of draws for which the probability $\mathbb {P}\left(P\right)$ of getting a black ball at least once is greater than $\frac{1}{2} \left(\mathbb {P}\left(P\right) > \frac{1}{2}\right)$.

Under the same conditions as in the previous problem — from a box containing 20 white balls and 2 black ones, $m$ balls are drawn without returning the ball to the box — determine the smallest value $x$ of the number of draws $m$ so that the probability $\mathbb {P}\left(P\right)$ of getting at least one white ball is more than $\frac{1}{2}$, if the ball is not put back.

How many times must a die be thrown so that the appearance of a 6 at least once has probability greater than 0.7? greater than 0.8? greater than 0.9?

A deck of cards is divided into two equal packs (26 cards each). Determine the probability $\mathbb {P}\left(P\right)$ that both packs will contain an equal (2) number of aces.

Under the same conditions as in the previous problem — a deck of cards is divided into two equal packs (26 cards each) — determine the probability $\mathbb {P}\left(P_{1}\right)$ that the number of red and black suits in each pack will be equal (13 each).

6 cards are drawn from a deck of cards. Determine the probability $\mathbb {P}\left(P\right)$ that among these cards representatives of all four suits will be present.

What is the smallest number $x_{0}$ of cards that must be drawn from the deck so that the probability $\mathbb {P}\left(P_{x_{0}}\right)$ that among them two cards of the same denomination will occur is greater than $\frac{1}{2}$?

20 people, among whom are 10 men and 10 women, are grouped in pairs. What is the probability $\mathbb {P}\left(P\right)$, if the grouping happens at random, that each of the 10 pairs consists of persons of different sex?

A group of 5 men and 10 women is split into 5 groups of 3 people each. What is the probability, under a random grouping, that each group will contain 1 man?

A telephone number consists of 5 digits. Determine the probability $\mathbb {P}\left(P\right)$ that all the digits are distinct (each digit can take 10 values).

Three urns contain white and black balls. The first contains 3 white balls and 1 black, the second urn 6 white and 4 black, the third — 9 white and 1 black. A ball is drawn from an urn chosen at random. Compute the probability that this ball is white, assuming that the choice of any of the urns is equally likely.

From an urn containing 3 white and 2 black balls, two balls are transferred to an urn containing 4 white and 4 black balls. Compute the probability of drawing a white ball from the second urn.

Three urns contain white and black balls. The first has 2 white and 3 black balls, the second 2 white and 2 black balls, the third 3 white and 1 black ball. A ball from the first urn is transferred to the second. After that a ball from the second urn is transferred to the third. Finally a ball from the third urn is transferred to the first.

Which distribution of black and white balls in the first urn appears most probable? Compute this probability.

Which appears more probable: the preservation of the original composition of the urn, or its change?

Under the conditions of the previous problem (problem 6150 — from an urn containing 3 white and 2 black balls, two balls are transferred to an urn containing 4 white and 4 black balls), determine the probability that the composition of balls in all the urns will not change as a result of the transfer.

Under the conditions of problem 6150 (from an urn containing 3 white and 2 black balls, two balls are transferred to an urn containing 4 white and 4 black balls), compute the probability that, in transferring balls from one urn to the other, a white ball was transferred twice and a black ball once. Indicate the probability that in this case the composition of the first urn will not change.

It is required to show that

in particular, if $A$ and $B$ are independent, then

Prove that if $Y = X + \delta$, where the probability that $X \leqslant 10$ equals 0.9, and the probability that $\left|\delta \right| \leqslant 1$ equals 0.95, then the probability $\mathbb {P}\left(P_{1}\right)$ that $Y \leqslant 9$, and the probability $\mathbb {P}\left(P_{2}\right)$ that $Y \leqslant 11$, satisfy the inequalities

whatever the dependence between $X$ and $\delta$ may be.

Prove that

Consider $n$ events $A_{1}, A_{2}, \ldots , A_{n}$; denote $p_{i} = \mathbb {P}\left(A_{i}\right)$; $p_{ik} = \mathbb {P}\left(A_{i} \text{ and } A_{k}\right)$, $p_{ikl} = \mathbb {P}\left(A_{i} \text{ and } A_{k} \text{ and } A_{l}\right)$ and so on. Show that the probability

From an urn containing balls numbered $1, 2, \ldots , N$, draws are made in succession. Each time after a draw the ball is returned to the urn. Compute the probability that, in $n$ successive draws, the numbers of the balls drawn in succession will come out in increasing order.

From the sequence of numbers $1, 2, \ldots , n$ a number $k$ is marked. What is the probability that two numbers chosen at random from this sequence will turn out to be, one less and the other greater, than $k$?

From the sequence of numbers $1, 2, \ldots , N$, $n$ numbers are chosen at random: $x_{1} < x_{2} < \cdots < x_{n}$. Find the probability that $x_{m} = M$.

From an urn containing balls numbered $1, 2, 3, \ldots , N$, $n$ successive draws are made. Each time after a draw the ball is returned to the urn. The numbers of the balls drawn are recorded in increasing order. Compute the probability that the number of the $m$-th ball will turn out to equal $M$.

Assuming in the previous problem that the numbers $M$ and $N$ are very large and their ratio equals $x$, find the limiting distribution law of the quantity $x$.

6 tickets with numbers $1, 2, \ldots , 6$, drawn in succession from a box, have equal probability of appearing in any order. It is required to determine (using the solution of problem 6157) the probability $\mathbb {P}\left(P\right)$ that the ordinal number of at least one of the tickets coincides with its own number. Solve the same problem if the number of tickets increases without bound.

Given $N$ tickets with numbers $1, 2, \ldots , N$, as in the previous problem. Determine the probability $\mathbb {P}\left(P\right)$ that, drawing $n < N$ of them in succession, at least once the ordinal number of the draw of a ticket coincides with its own number. Consider the case $N = 10$, $n = 5$.

From a vessel containing $m$ white and $n$ black balls, balls are drawn in succession and not returned to the urn. The operation continues until a white ball appears. Find the distribution law of the number of black balls drawn.

Assuming in the previous problem that the quantities $m$, $n$ are very large, establish the limiting distribution law. The ratio $n : (m+n) = a$.

From a vessel containing $m$ white and $n$ black balls, a ball is drawn and returned to the vessel. The drawing continues until a white ball appears. Compute the probability of drawing a black ball exactly $k$ times.

Assuming in problem 6167 that the quantities $m$ and $n$ increase without bound and the ratio $n : (m+n)$ tends to $a$, establish the limiting distribution law.

From a vessel containing 2 white and 4 black balls, two people draw balls alternately. After a draw the ball is returned to the vessel each time. The drawing stops when a white ball appears.

Compute the probability that each of the participants draws a white ball first, if the ball is drawn no more than $2k$ times.

From a vessel containing 2 white and 4 black balls, two people draw a ball alternately. Compute the probability that each of the participants draws a white ball first.

Two people play a game by tossing a coin. The winner is considered to be whoever first tosses heads. How many times greater is the probability of winning for the one who starts, if the coin tossing can be continued indefinitely?

A vessel contains $N$ tickets with distinct numbers. From the vessel, $n$ tickets are drawn $m$ times, being returned each time. Compute the probability that $k$ tickets will not appear. Compute the probability that all the tickets will appear.

A vessel contains $N$ tickets with distinct numbers. From the vessel, $n$ tickets are drawn $k$ times and returned after each draw. Establish the distribution law for the number of distinct numbers that appeared over the $k$ draws.

Two boxes are given: the first contains 2 white, 3 red, and 20 black balls, the second — 8 white, 15 red, and 2 black balls. After drawing a ball first from one box and then from the other, the balls drawn turned out to be first black, then red.

Determine the probability that the draws were made first from the first box and then from the second, assuming both sequences a priori equally likely.

Event $A$ has probability $\mathbb {P}\left(P\right)$ of occurring at any moment of the time interval $T$. Determine its probability $p$ of occurring during the remaining time $T-t$, if by the time $t < T$ event $A$ has not yet occurred.

Given that the probability that two twins are of the same sex is twice the probability of being of different sex, and that, in general, the probability of a boy being born equals 0.51, determine the probability that the other twin is a boy, given that it has been established that the first of them is a boy.

What is the probability $p_{0}$ that among 1000 light bulbs there is not a defective one, if of 100 taken at random all turned out to be sound? Solve the problem under the assumption that the number of bad light bulbs cannot exceed 5 per thousand and that all values (0, 1, 2, 3, 4, 5) of the number of defective bulbs are a priori equally possible.

A white ball is dropped into a vessel containing $n$ balls. What is the probability of drawing a white ball from this vessel? All assumptions about the original composition of the balls are taken as equally possible.

From a vessel containing $n$ balls of unknown color, a ball is drawn and turns out to be white. Compute the probability that a newly drawn ball will also be white. All assumptions about the original composition of the balls in the vessel are taken as equally possible.

From a vessel containing $n$ balls of unknown color, a ball is drawn and turns out to be white. After this ball is returned to the urn, a ball is drawn again. Determine the probability that this ball will be white. The original composition of the vessel is assumed equally probable.

From a vessel containing $N$ balls, $m+n$ balls are drawn, among which $m$ balls are white. Establish the probability that before the withdrawal there were $M$ white balls in the vessel.

From a heap containing on its surface $n$ pieces of oxidized ore and $n'$ pieces of sulfide ore, $m$ samples are selected at random and sent to the laboratory for analysis. On opening the box of samples it turned out that the first $k$ of the pieces drawn belong to the oxidized ores.

What is the probability that the next piece will also belong to the same ore? In solving the problem, assume that any $m$ of the $n+n'$ pieces on the surface of the heap could have been selected into the box with equal probability.

A vessel contains white and black balls. The total number $N$ of balls in the vessel is known, but neither the number of white nor the number of black balls is known.

In $(m+n)$ draws of a ball from the vessel, a white ball appeared $m$ times and a black one $n$ times.

What composition of balls in the vessel appears most probable? All assumptions about the original composition of the vessel are equally probable.

Denoting by $\mathbb {P}\left(P_{i,n}\right)$ the probability of event $A$ occurring $i$ times in $n$ independent trials, if the probabilities of $A$ are successively $p_{1}, p_{2}, \ldots , p_{n}$, prove the inequalities

from which it follows that $\mathbb {P}\left(P_{i,n}\right)$ increases with $i$ up to some maximum value, after which it begins to decrease.

Example. $p_{1} = \frac{1}{2}$, $p_{2} = \frac{1}{3}$, $p_{3} = \frac{1}{4}$, $p_{4} = \frac{1}{5}$, $p_{5} = \frac{1}{6}$.

Given 3 pairwise independent events which, however, cannot all 3 occur together. Assuming they all have the same probability $x$, determine the largest possible value of $x$.