Basic probability, discrete and continuous distributions, and related combinatorial arguments.
These practice problems are elementary probability: events, counting, and the rules that combine probabilities. Sample spaces are finite, or they are geometric models measured by length and area. You will compute a probability from equally likely outcomes, from a region, and from conditional information. The write-ups keep the sample space visible, so the counting or the diagram can be checked before the arithmetic.
Random Events and Classical Probability are the chapters to open first: outcomes, equally likely cases, and counting. Geometric Probability replaces counting with length, area, or volume when the experiment is "pick a point at random." Properties of Probability and Addition and Multiplication Rules cover complements, inclusion, and independent trials. Conditional Probability and Independence is the chapter for updating a probability given new information and for checking whether two events are independent.
Most of these problems do not need calculus. A geometric model is easier if area is already familiar from Applications of Definite Integrals. Use Easy problems for classical counting, then the conditional chapter, which is the harder part of the set. Limits and Series are separate calculus topics and are not required here.
Problem 1
HardRandom Events
Let $A$, $B$ and $C$ be random events. Explain the meaning of the equalities
$1^{\circ}\quad ABC = A; \qquad\qquad 2^{\circ}\quad A + B + C = A.$
From a table of random numbers, two numbers are chosen at random. Let $A$ denote the event that (at least) one of these numbers is prime, and $B$ the event that (at least) one of them is even. What do the events $AB$ and $A+B$ represent?
A target consists of ten concentric circles with radii $r_k$ $(k = 1, 2, \dots, 10)$, where $r_1 \lt r_2 \lt \dots \lt r_{10}$. Event $A_k$ denotes a hit in the circle of radius $r_k$ $(k = 1, 2, \dots, 10)$. What do the events
$B = \sum_{k=1}^{6} A_k \qquad\text{and}\qquad C = \prod_{k=1}^{10} A_k$
mean?
A square $D$ contains two regions $D_1$ and $D_2$ with areas $S_1$ and $S_2$. A point is thrown into $D$ and can fall into any part of the square with equal probability. Let event $A_k$ denote that the point falls into region $D_k$ $(k = 1, 2)$. Are the events $A_1$ and $A_2$ equally likely?
Three devices are being tested. Let $A$ be the event that exactly one of the three devices is defective, and let $B$ be the event that the devices are functional. What do the events $A+B$ and $AB$ represent?
From a table of random numbers, one number is selected at random. If the selected number is divisible by $5$, this is event $A$; if the number ends with zero, this is event $B$. What does the event $A\setminus B$ mean?
If at least one of four products is defective, this is event $A$; if no fewer than two products are defective, this is event $B$. What do the complementary (opposite) events $\overline{A}$ and $\overline{B}$ mean?
If $A$, $B$ and $C$ are random events, simplify the expressions:
$1^{\circ}\ (A+B)(B+C); \qquad 2^{\circ}\ (A+B)(A+\overline{B}); \qquad 3^{\circ}\ (A+B)(A+\overline{B})(\overline{A}+B).$
Verify the correctness of the following equalities:
\[\begin{aligned}
&1^{\circ}\ \overline{\overline{A}\,\overline{B}}=A+B; \qquad 2^{\circ}\ \overline{\overline{A}+\overline{B}}=AB; \qquad 3^{\circ}\ A+B=AB+A\triangle B;\\
&4^{\circ}\ \overline{A\triangle B}=AB+\overline{A}\,\overline{B}; \qquad 5^{\circ}\ A\triangle B=\big(\overline{A\overline{B}}\big)\triangle\big(\overline{\overline{A}B}\big);\\
&6^{\circ}\ \overline{\textstyle\bigcup_{i=1}^{n} A_i}=\textstyle\bigcap_{i=1}^{n}\overline{A_i}; \qquad 7^{\circ}\ \overline{\textstyle\bigcap_{i=1}^{n} A_i}=\textstyle\bigcup_{i=1}^{n}\overline{A_i}.
\end{aligned}\]
Here $A\triangle B = A\overline{B}+\overline{A}B$ is the symmetric difference of $A$ and $B$.
Two chess players play one game. If the first player wins, this is event $A$; if the second player wins, this is event $B$. What event should be added to this set of events to obtain a complete system of events?
Among the students attending a probability theory class, we randomly select one. Let event $A$ mean that the selected student is male, event $B$ that the student does not smoke, and event $C$ that the student lives in a student dormitory.
$1^{\circ}$ Describe the event $AB\overline{C}$.
$2^{\circ}$ Under what conditions does the identity $ABC=A$ hold?
$3^{\circ}$ When is the relation $\overline{C}\subseteq B$ correct?
$4^{\circ}$ Explain when the equality $\overline{A}=B$ holds. Does it hold when all the male students present do not smoke?
Prove that for arbitrary events $A$ and $B$ the following relations are equivalent:
$\text{(i)}\ A\subseteq B; \qquad \text{(ii)}\ \overline{A}\supseteq\overline{B}; \qquad \text{(iii)}\ A+B=B; \qquad \text{(iv)}\ A\overline{B}=\varnothing.$
Prove that the events
$1^{\circ}\ (A+B)(A+\overline{B})+(\overline{A}+B)(\overline{A}+\overline{B}) \qquad\text{and}\qquad 2^{\circ}\ (A+B)(\overline{A}+B)+(A+\overline{B})(\overline{A}+\overline{B})$
are certain events.
It is known that $P(AB)=P(A)P(B)$ (i.e. the events $A$ and $B$ are independent), $C\supset AB$ and $\overline{C}\supset\overline{A}\,\overline{B}$. Prove that $P(AC)>P(A)P(C)$.
It is known that the simultaneous occurrence of the events $A_1$ and $A_2$ necessarily implies the occurrence of the event $A$. Prove that
$P(A) > P(A_1) + P(A_2) - 1.$
A worker has produced $n$ articles. Let event $A_i$ $(i=1,2,\dots,n)$ mean that the $i$-th article produced is defective. Write out the following events:
$1^{\circ}$ all products are non-defective;
$2^{\circ}$ at least one product is defective;
$3^{\circ}$ exactly one product is defective;
$4^{\circ}$ at most two products are defective;
$5^{\circ}$ at least two products are non-defective;
$6^{\circ}$ exactly two products are defective.
Let $A_n$ denote the event that the event $A$ occurs on the $n$-th trial, and let $B_{n,m}$ denote the event that in the first $n$ trials the event $A$ occurs exactly $m$ times.
$1^{\circ}$ Express $B_{4,2}$ in terms of the $A_i$.
$2^{\circ}$ What is the meaning of the event $\displaystyle B_m=\bigcap_{n=1}^{\infty}\Big(\bigcup_{k=0}^{m}B_{n,k}\Big)$?
$3^{\circ}$ Are the relations $\displaystyle\bigcap_{n=1}^{\infty}A_n\subseteq\overline{B}$ and $\displaystyle\bigcap_{n=1}^{\infty}\overline{A_n}\subseteq B$ correct, where $B=\bigcup_{m}B_m$?
In a set $\Omega$ of points, $n$ subsets $A_i$ $(i=1,2,\dots,n)$ are distinguished. For any set $C$ define its characteristic function $\chi_C$ by $\chi_C(\omega)=1$ if $\omega\in C$ and $\chi_C(\omega)=0$ otherwise. Prove that using the $A_i$ one can construct sets $B_k$ $(k=1,2,\dots,2^n)$ such that for any bounded function
$F(\omega)=F\big(\chi_{A_1}(\omega),\dots,\chi_{A_n}(\omega)\big)$
there are constants $C_k$ with
$F(\omega)=\sum_{k} C_k\,\chi_{B_k}(\omega).$
A box contains ten balls: three white and seven black. One ball is drawn at random from the box. Determine the probability that the drawn ball will be: $1^{\circ}$ white; $2^{\circ}$ black.
A cube with all its faces coloured is divided into one thousand cubes of equal size. The cubes obtained in this way are well mixed. Determine the probability that a randomly drawn cube will have exactly two coloured faces.
In a batch of $n$ products there are $k$ defective items. Determine the probability that among $m$ randomly chosen products exactly $l$ will be defective.
A lottery has been released for sale where the total value of tickets is $n$ dinars. The entire prize amount is distributed among $m$ tickets. Find the expected value (mathematical expectation) of the prize for one ticket.
From a batch of products, among which there are $n$ non-defective and $m$ defective, $s$ specimens are taken for inspection. It was observed that the first $k$ of the $s$ specimens taken are non-defective. Determine the probability that the next specimen will also be non-defective.
Determine the probability that for a randomly chosen integer $N$ the following will end in the digit $1$:
$1^{\circ}$ its square;
$2^{\circ}$ its fourth power;
$3^{\circ}$ the number obtained by multiplying $N$ by an arbitrary (randomly chosen) integer.
A lottery has released $n+m$ tickets for sale, of which only $n$ are winning tickets. If someone bought $k$ tickets, find the probability that among them there will be exactly $s$ winning tickets.
In a block of ten cards (tickets) there are five cards worth $100$ dinars, three cards worth $300$ dinars and two cards worth $500$ dinars. Three cards are drawn at random. What is the probability that:
$1^{\circ}$ at least two of the drawn cards have the same value;
$2^{\circ}$ the total value of the three drawn cards is $700$ dinars?
From a box containing $n$ banknotes, at least one banknote is drawn. Determine the probability that an even number of banknotes will be drawn, if drawing any set of banknotes from the given ones is equally likely.
A deck of $36$ cards is randomly divided into two equal parts. What is the probability that both parts will contain the same number of red and black cards?
A deck of $52$ cards consists of four suits of $13$ cards each. The deck is thoroughly shuffled, so that every card is equally likely to be drawn. Six cards are drawn. Describe the sample space and:
$1^{\circ}$ find the probability that the king of spades is among the drawn cards;
$2^{\circ}$ find the probability that all four suits are represented among the drawn cards;
$3^{\circ}$ what is the minimum number of cards that must be drawn so that the probability that two of them have the same rank is greater than $\tfrac12$?
Find the probability that when a deck of $52$ cards is dealt to four players, the first player receives exactly $n$ pairs "ace and king of the same suit".
In a closet there are $n$ pairs of shoes. From the closet, $2r$ shoes are chosen at random ($2r\lt n$). What is the probability that among the chosen shoes:
$1^{\circ}$ there is no complete pair;
$2^{\circ}$ there is exactly one complete pair;
$3^{\circ}$ there are exactly two complete pairs?
A box contains $n$ white and $m$ black balls ($m\lt n$). All balls are drawn one by one without replacement. Let $M(k)$ be the number of black balls and $N(k)$ the number of white balls among the first $k$ drawn. Find the probability $p$ that $M(k)\lt N(k)$ for every $k=1,2,\dots,n+m$.
Banach's matchbox problem. A mathematician carries two boxes of matches, each initially containing $n$ matches. Each time he needs a match, he chooses one of the boxes at random. Find the probability that when he first finds a box empty, the other box contains exactly $r$ matches $(r=0,1,\dots,n)$.
Let $\varphi(n)$ denote the number of natural numbers $\le n$ that are relatively prime to $n$. Prove that
$\varphi(n)=n\prod_{p\mid n}\left(1-\frac1p\right),$
where the product is taken over all prime divisors $p$ of $n$.
Consider a physical system of $r$ indistinguishable particles. In statistical mechanics the phase space is divided into a large number $n$ of cells, and each particle falls into one of the cells. The state of the system is then a distribution of the $r$ particles among the $n$ cells, uniquely determined by the numbers $0\le m_i\le r$ $(i=1,\dots,n)$, where $m_i$ is the number of particles in cell $i$. Photons, atomic nuclei and atoms containing an even number of elementary particles obey Bose–Einstein statistics, in which only the different distributions are considered and each is assigned the same probability. Find this probability.
Let $r$ particles be distributed among $n$ cells according to Bose–Einstein statistics (see Problem 58).
$1^{\circ}$ Prove that the probability that exactly $k$ particles are in a fixed cell is
$q_k=\frac{\binom{n+r-k-2}{r-k}}{\binom{n+r-1}{r}}.$
$2^{\circ}$ Show that $q_0\gt q_1\gt q_2\gt\dots$
$3^{\circ}$ Prove that if $n$ and $r$ grow without bound while the average number of particles per cell $\frac rn$ tends to $\lambda\lt\infty$, then
$q_k\to\frac{\lambda^k}{(1+\lambda)^{k+1}}$
(the right side is the geometric distribution).
Given $n$ particles, each of which can be found with equal probability $\frac1N$ in each of $N$ $(N\gt n)$ cells. Find the probability that:
$1^{\circ}$ exactly one particle falls into each of $n$ specified cells;
$2^{\circ}$ exactly one particle falls into each of some $n$ cells.
A beam of $k$ particles is caught by $n$ counters placed in a row, which register particles. Each particle can fall on any counter with equal probability. What is the probability that exactly $r$ counters register particles?
Show that
$\varrho(A,B)=P(A\triangle B)$
satisfies all axioms of a metric space except the identity axiom ($\varrho(A,B)=0$ need not imply $A=B$), and that for arbitrary events $A,B,C$
$\varrho(A,B)+\varrho(B,C)\ge\varrho(A,C).$
Let us agree that for events $A_i,A_j,A_k$ property $b_{ijk}$ holds for $A_i, A_j, A_k$ if
$1^{\circ}\ \ A_i\,\overline{A_j}\,A_k = V, \qquad 2^{\circ}\ \ \overline{A_i}\,A_j\,\overline{A_k} = V,$
where $V$ denotes the impossible event. Prove:
1) if property $b_{ijk}$ holds, then $\varrho(A_i,A_j)+\varrho(A_j,A_k)=\varrho(A_i,A_k)$, where $\varrho(A,B)=P(A\triangle B)$ (Problem 63);
2) if $P(A)=1$ implies $A=E$ (the certain event), then the converse also holds.
Let $A^*=(A_1,\dots,A_n)$ and $B^*=(B_1,\dots,B_n)$ be two systems of sets with $A_{j+1}\supseteq A_j$ and $B_{j+1}\supseteq B_j$ $(j=1,\dots,n-1)$, with $A_nB_n=V$, and let $C$ be a set with $A_nC=B_nC=V$. The sequence $L^*=(L_1,\dots,L_n)$ with
$L_i=A_i+B_{n-i+1}+C$
is called linearly ordered. Prove that if $i\lt j\lt k$, then property $b_{ijk}$ holds for $L_i,L_j,L_k$ (see Problem 64).
Prove, using Problems 64 and 66, that for a linearly ordered sequence of sets $L^*$ and $i\lt j\lt k$
$\varrho(L_i,L_j)+\varrho(L_j,L_k)=\varrho(L_i,L_k).$
Prove that if $R^*=(R_1,\dots,R_n)$ is a sequence of sets such that property $b_{ijk}$ holds for $R_i,R_j,R_k$ for every $i\lt j\lt k$, then $R^*$ is a linearly ordered sequence of sets (see Problem 66).
A metal coin of diameter $2r\lt a$ is thrown onto an infinite chessboard with squares of side $a$. Find the probability that:
$1^{\circ}$ the whole coin falls inside one of the squares;
$2^{\circ}$ the coin intersects exactly one side of a square.
On a section of magnetic tape of length $200$ m, information is recorded on an interval of $20$ m. On another interval, independently of the first, similar information is recorded. Determine the probability that there will be continuous recording on the interval from $60$ to $80$ m, if the starting points of both recordings are equally likely anywhere from $0$ to $180$ m.
Two signals arrive at a receiver, and every moment of a time interval of length $T$ is equally likely for the arrival of each signal. The receiver is jammed if the difference between the arrival times of the two signals is less than $\tau$ $(\tau\lt T)$. Determine the probability that the receiver will be jammed.
A point is thrown at random inside a circle of radius $r$; the probability of falling into a region inside the circle is proportional to its area. Determine the probability that:
$1^{\circ}$ the point is at distance at most $d$ $(d\lt r)$ from the centre;
$2^{\circ}$ the smaller angle between a given direction and the line joining the centre to the point does not exceed $\alpha$.
At some point $C$ of a telephone line $AB$ of length $L$ a break has occurred. Find the probability that the distance from $C$ to $A$ is not less than $l$.
A point $M$ is thrown at random into the square with vertices $(0,0),(0,1),(1,0),(1,1)$; let $(u,v)$ be its coordinates. The probability that the point falls into a region of the square is proportional to the area of the region.
$1^{\circ}$ Prove that for $0\lt x,y\lt1$: $\ P(u\lt x,\ v\lt y)=P(u\lt x)\,P(v\lt y)=xy$.
$2^{\circ}$ For $0\lt z\lt1$ find: 1) $P(|u-v|\lt z)$; 2) $P(uv\lt z)$; 3) $P(\min(u,v)\lt z)$; 4) $P(\max(u,v)\lt z)$; 5) $P\big(\tfrac12(u+v)\lt z\big)$.
In a circle of radius $R$, chords are drawn parallel to a given direction. What is the probability that the length of a randomly chosen chord is not greater than $r$, if all positions of the intersection point of the chord with the diameter perpendicular to the given direction are equally likely?
A horizontal plane is ruled by parallel lines at distance $2a$ from each other. A convex contour whose greatest diameter is less than $2a$ is thrown on the plane at random. Find the probability that the contour intersects one of the lines.
A rectangular grid with cylindrical holes of radius $r$ is given; the distances between the axes of neighbouring holes are $a$ and $b$. Determine the probability that a ball of radius $d$ falls into a hole in one throw without aiming, if the ball falls perpendicular to the plane of the grid.
A point $M$ is thrown at random into a right triangle $ABC$ with legs $AB=l$ and $BC=k$. Determine the joint distribution of the length $h$ of the perpendicular dropped from $M$ to $AB$ and the angle $\alpha=\angle MAB$, i.e. for all $x$ and $y$ find the probability that the events $(h\lt x)$ and $(\alpha\lt y)$ occur simultaneously.
Inside an ellipse with semi-axes $a=100$ cm and $b=10$ cm a rectangle is placed symmetrically with respect to the centre of the ellipse, with its longer side parallel to the axis $a$; its sides are $10$ cm and $3$ cm. In addition, four circles of radius $4.3$ cm are placed inside the ellipse, not intersecting the ellipse, the rectangle or each other. Determine the probability that:
$1^{\circ}$ a random point, whose position is equally likely anywhere inside the ellipse, lies in one of the circles;
$2^{\circ}$ a circle of radius $5$ cm drawn around that point as centre intersects at least one side of the rectangle.
Two points $M$ and $N$ are chosen at random on a segment $AB$ of length $l$. Determine the probability that none of the three segments obtained is longer than $a$, where $l\gt a\gt\frac l3$.
Two ships must arrive at the same berth. The arrival times of both ships are independent and equally likely at any moment of the day (24 hours). Find the probability that one of the ships will have to wait for the berth to become free, if the first ship stays at the berth for one hour and the second for two hours.
Two persons are equally likely to arrive at a certain place at any moment of a time interval of length $T$. Determine the probability that one person will wait for the other no longer than $t$.
Two points are chosen at random on the surface of a sphere of radius $R$. What is the probability that the arc of the great circle through these two points subtends a (central) angle less than $\alpha$ $(\alpha\lt\pi)$?
An artificial Earth satellite moves in an orbit between $60^\circ$ north and $60^\circ$ south latitude. Assuming that the satellite can fall on the Earth's surface with equal probability at any point between these parallels, find the probability that it will fall north of $30^\circ$ north latitude.
A plane is ruled by parallel lines at distance $d$ from each other. Find the probability that a needle of length $l$ $(l\lt d)$ thrown at random crosses one of the lines (Buffon's problem).
Determine the probability that the roots of the equations
$1^{\circ}$ the quadratic $x^2+2ax+b=0$;
$2^{\circ}$ the cubic $x^3+3ax+2b=0$
are real, if the coefficients are equally likely to lie anywhere in the rectangle $|a|\le n$, $|b|\le m$. What is the probability that under these conditions the roots of the quadratic equation are positive?
In a plane, a point $A$ and the centre $B$ of a circle of radius $R$ move independently along straight lines with constant speeds $u$ and $v$. At a given moment $AB=r$ $(r\gt R)$, and the angle between the line $AB$ and the velocity $\vec v$ of the circle is $\beta$. Assuming that all directions of motion of $A$ are equally likely, find the probability that the point $A$ enters the circle.
Bertrand's paradox. A chord of a circle is chosen at random. Find the probability that its length is greater than the side of the inscribed equilateral triangle.
When the voltage is doubled, the electrical circuit may break because of the failure of one of three elements connected in series; the elements fail with probabilities $0.3$, $0.4$ and $0.6$ respectively. Determine the probability that the circuit will not break. By how much will this probability change if the first element is removed?
Determine the probability that a randomly chosen product is first-class, if it is known that $4\%$ of the products are defective and that $75\%$ of the non-defective products satisfy the conditions of first class.
A batch of one hundred products is subjected to partial inspection: five products are inspected, and if at least one defective product appears among them, the whole batch is rejected. What is the probability that the batch will be rejected if it contains $5\%$ defective products?
A lottery has $1000$ tickets: one ticket wins $500$ thousand, $10$ tickets win $100$ thousand each, $50$ tickets win $20$ thousand each, $100$ tickets win $5$ thousand dinars each, and the remaining tickets win nothing. Someone bought one ticket. What is the probability of winning at least $20$ thousand dinars?
Two shooters, who hit the target with probabilities $0.7$ and $0.8$ respectively, each fire one shot. Determine the probability that the target will be hit.
The probability that the $k$-th block of a computer fails during a time $T$ is $p_k$ $(k=1,2,\dots,n)$. Determine the probability that during this time at least one block fails, if the blocks work independently.
How many numbers must be taken from a table of random numbers so that we can be confident with probability $0.9$ that at least one even number will be among them?
A box contains two white and three black balls. Two balls are drawn one after another (without replacement). Find the probability that both balls are white.
The same conditions as in the previous problem (two white and three black balls, two balls drawn), but the first ball is returned to the box before the second draw.
A device working during a time $t$ consists of three parts, each of which may fail during this time independently of the others. If any part fails, the whole device stops working. The probabilities that the parts do not fail during time $t$ are $p_1=0.8$, $p_2=0.9$, $p_3=0.7$. Find the reliability of the whole device.
Show that if the conditional probability $P(A\mid B)$ is greater than the unconditional probability $P(A)$, then the conditional probability $P(B\mid A)$ is also greater than the unconditional probability $P(B)$.
A student came to an exam knowing only $20$ of the $25$ possible questions. The examiner gave the student three questions. Using conditional probability, find the probability that the student knows all three questions. Determine the same probability using the classical definition of probability.
To find a book, a student decided to visit three libraries. For each library it is equally likely that the book is or is not in its collection; if the library has the book, it is equally likely that the book is or is not checked out by other readers. The libraries acquire books independently of each other. What is more likely — that the student will find the book or will not?
Six cards, each with one letter, form the word "kareta". The cards are shuffled and then drawn one by one at random. What is the probability that the letters drawn in order form the word "raketa"?
Two airplanes bomb the same target one after the other, each dropping one bomb per pass. Each airplane has three bombs. The probability that the first airplane hits the target with one bomb is $p_1=0.3$, for the second $p_2=0.4$. If one bomb hits the target, the target is destroyed and the bombing stops. Find the probability that the target will be destroyed before all the bombs of the airplanes are used up.
Let $A_1,A_2,\dots,A_n$ be pairwise independent events. Are these events always independent as a whole (mutually independent), i.e. is the conditional probability of any one of them, given that any other events of this set have occurred, always equal to its unconditional probability?
In a group of $2n$ people there are equally many men and women. They take seats at a table at random. Determine the probability that no two people of the same gender sit next to each other.
A box contains $n$ balls numbered from $1$ to $n$. The balls are drawn at random without replacement. What is the probability that in each of the first $k$ draws the number on the ball coincides with the number of the draw?
A box contains two balls, one white and one black. Balls are drawn one at a time until the black ball appears; each time a white ball is drawn, it is returned to the box together with two more white balls. Find the probability that the black ball is not drawn in the first $50$ trials.
In an election between two candidates, $n$ ballots for the first candidate and $m$ for the second were placed in the ballot box, $n\gt m$. What is the probability that, when the ballots are drawn one by one, the number of votes for the first candidate will always be greater than the number of votes for the second?
In a queue for tickets costing $5$ dinars each, $n+m$ people are waiting: $n$ of them have $5$-dinar bills and the remaining $m\lt n$ have $10$-dinar bills. Each person buys one ticket. What is the probability that nobody will have to wait for change? (The cashier has no money at the start.)
$1^{\circ}$ Prove that if $P(A)=0.9$ and $P(B)=0.8$, then $P(AB)\ge0.7$.
$2^{\circ}$ Prove that $P(A_2\mid A_1)\ge1-\dfrac{P(\overline{A_2})}{P(A_1)}$.
Using the formula $P(B)=P(A)P(B\mid A)+P(\overline A)P(B\mid\overline A)$, show that if the random events $A$ and $B$ are independent, then the events $\overline A$ and $B$ are also independent.
Determine the probability that a batch of one hundred products, five of which are defective, will be accepted when a randomly selected half of the batch is tested, if the acceptance condition allows at most one defective item among the fifty tested.
Three ammunition warehouses are bombed, and only one bomb is dropped. The probability that it falls on the first warehouse is $0.01$, on the second $0.008$, on the third $0.025$. If the bomb falls on any one of the warehouses, it destroys all three. Find the probability that the warehouses will be destroyed.
The probability that an event $A$ occurs during an interval of length $T$ is $p$, and the event is equally likely to occur at any moment of this interval. It is known that during the time $t\lt T$ the event has not occurred. Determine the probability that the event $A$ will occur in the remaining part of the interval.
Three shots are fired at the same target. The probabilities of hitting the target with the first, second and third shot are $p_1=0.4$, $p_2=0.5$, $p_3=0.7$. Determine the probability that there will be exactly one hit in the three shots.
An air duel takes place between a fighter and a bomber. The fighter fires first, one burst, and shoots down the bomber with probability $0.2$. If the bomber is not shot down, it returns fire and shoots down the fighter with probability $0.3$. If the fighter is not shot down, it attacks again from closer range and shoots down the bomber with probability $0.4$. Find the probability that in the duel $1^{\circ}$ the bomber is shot down; $2^{\circ}$ the fighter is shot down.
An airplane under fire consists of three parts of different vulnerability. To destroy the airplane it is sufficient to hit the first part once, or the second part twice, or the third part three times. The conditional probability that a hit falls on a given part is proportional to its area; the parts occupy $0.1$, $0.2$ and $0.7$ of the total area. Suppose the airplane was hit by two shells. Find the probability that it is destroyed.
For an airplane to be destroyed by a shell exploding at some distance from it, it is necessary that both engines are destroyed by shell fragments or that the pilot is killed. If a shell explodes at distance $r$ from the airplane, the probability that both engines are destroyed is $0.2$, and the probability that the pilot is killed is $0.3$. Find the probability that the airplane is knocked out by the explosion.
A circular target consists of three zones: I, II and III. The probability of hitting the first zone with one shot is $0.15$, the second $0.23$, the third $0.17$. Find the probability of a miss.
A worker tends three looms that work independently of each other. The probability that a loom will not require the worker's attention during one hour is $0.9$ for the first, $0.8$ for the second and $0.7$ for the third. Find the probability that during one hour at least one loom will not require the worker's attention.
Balls are drawn one by one at random from a box containing $n$ white, $m$ black and $l$ red balls: $1^{\circ}$ without replacement; $2^{\circ}$ with replacement after each draw. Determine in both cases the probability that a white ball is drawn before a black ball.
Someone wrote $n$ letters to different people, sealed them in envelopes, and then wrote the $n$ different addresses on the envelopes at random. Determine the probability that at least one letter will be correctly addressed.
As a result of an experiment one of four mutually exclusive events can occur, with probabilities $0.012$, $0.010$, $0.006$ and $0.002$ respectively. Determine the probability that one of these events occurs as a result of the experiment.
A square is divided into $n^2$ equal squares. A ball is thrown into it; the probability that it falls into the small square in the $i$-th horizontal and $j$-th vertical strip is $p_{ij}$, with $\sum_{i,j=1}^np_{ij}=1$. Determine the probability that the ball falls into the $k$-th horizontal strip.
Determine the probability that a randomly chosen integer is not divisible $1^{\circ}$ by 2 nor by 3 (by neither of them); $2^{\circ}$ by 2 or by 3 (i.e. is not divisible by 2, or is not divisible by 3).
The probability of obtaining a numbered ticket (six digits) in which the sum of the first three digits equals the sum of the last three digits is $0.05525$. What is the probability that among two tickets there is such a ticket, if the two tickets $1^{\circ}$ have consecutive numbers; $2^{\circ}$ are independent of each other?
It is known that $P(x\lt10)=0.9$ and $P(|y|\lt1)=0.95$. Prove that, whatever the dependence between $x$ and $y$, for $z=x+y$ the inequalities $P(z\lt11)\ge0.85$ and $P(z\lt9)\le0.95$ hold.
Players $a$ and $b$ toss a coin. The first player wins if heads appears on the first, second or third toss; otherwise the second player wins. Determine the probability that the second player wins.
Two players alternately toss a coin. The one who first gets heads wins. Determine the probability that the first player wins and the probability that the second player wins.