Elementary Probabilities

Let $\Omega$ denote the totality of students at the University of Athens and $A_{1}, A_{2}, A_{3}, A_{4}$, the sets of freshmen, sophomores, juniors, and seniors, respectively. Moreover, let $F$ denote the set of female students and $C$ the set of Cypriot students. Express in words each of the following sets:

Give the simplified forms of the sets:

Express each of the following events in terms of the events $A$, $B$, and $C$, and the operations of complementation, union, and intersection:

Let $\Omega$ denote the sample space corresponding to the chance experiment of tossing a coin three times. Let $A$ be the event that heads appear exactly twice, let $B$ be the event that at least two heads appear, and let $C$ the event that heads appear when tails have appeared at least once.

In each of the exercises 5--8, give the number of points in the sample space $\Omega$ and the points of the events defined therein.

A family has 4 children. Let the events: $A$: boys and girls alternate, $B$: the first and fourth child are boys, $C$: as many boys as girls, and $D$: three successive children of the same sex.

A salesman is arranging his schedule for visiting each of three towns $a$, $b$, $c$, twice. $A$: the first and last visit are in $a$.

Certain diseases cause cell disorders which can be distinguished into four categories. Blood tests are conducted on each of 4 patients and in each case the category is noted. Let $A$: the patients belong to the same category, $B$: two patients belong to the same category.

An elevator carries two persons and stops at three floors. Let $A$: they get off at different floors, $B$: one gets off at the first floor.

In how many ways can a lady having 10 dresses, 5 pairs of shoes, and 2 hats be dressed?

In how many ways can we place in a bookcase two works each of three volumes and two works each of four volumes, so that the volumes of the same work are not separated?

In how many ways can $r$ objects be distributed to $n$ persons if there is no restriction on the number of objects that a person may receive?

In how many ways can 5 boys and 5 girls be seated around a table so that no 2 boys sit next to each other?

Eight points are chosen on the circumference of a circle. How many chords can be drawn by joining these points in all possible ways? If the 8 points are considered vertices of a polygon, how many triangles and how many hexagons can be formed?

In how many ways can $n$ persons be seated around a table if 2 arrangements are regarded as the same when each person has the same right and left neighbors? At a dinner at which the $n$ persons are seated randomly, what is the probability that a specific husband is seated next to his wife?

In how many ways can 20 recruits be distributed into 4 groups each consisting of 5 recruits? In how many ways can they be distributed into 4 camps, each camp receiving 5 recruits?

Using 7 consonants and 5 vowels, how many words consisting of 4 consonants and 3 vowels can we form?

Show that for every $x$ and for every positive integer $n$:

1. $\binom {x}{n-1} + \binom {x}{n} = \binom {x+1}{n}$;

2. $\binom {x}{n} = \binom {x-1}{n-1} + \binom {x-2}{n-1} + \cdots + \binom {n-1}{n-1}$.

If $n$, $m$, $r$, are positive integers prove that

For every integer $n \geq 2$, show the following:

1. $1 - \binom {n}{1} + \binom {n}{2} + \cdots + (-1)^{n}\binom {n}{n} = 0$;

2. $\binom {n}{1} + 2\binom {n}{2} + 3\binom {n}{3} + \cdots = n2^{n-1}$;

3. $2 \cdot 1 \cdot \binom {n}{2} + 3 \cdot 2 \cdot \binom {n}{3} + 4 \cdot 3 \cdot \binom {n}{4} + \cdots = n(n-1)2^{n-2}$.

Using Pascal's triangle, show that for every $a$ and positive integers $r$, $n$:

1. $\displaystyle \sum_{k=0}^{n} \binom {a-k}{r} = \binom {a+1}{r+1} - \binom {a-n}{r+1}$;

2. $\displaystyle \sum_{k=0}^{n} (-1)^{k}\binom {a}{k} = (-1)^{n}\binom {a-1}{n}$.

Using the result of Exercise 18, show that

Using the result of Exercise 21, prove that

Using the binomial theorem, show by induction that

As in Exercise 23, show that for positive integers $r$ and $n$

Show that this is a special case of Problem 20(ii).

In the expansion of $(1+x)^{n}$ where $x > 0$ and $n$ is a positive integer, let $a_{k}$ be the term containing $x^{k}$. Find the values of $k$ for which $a_{k}$ becomes maximum. What is the maximum term of $(1+e)^{100}$?

Given $\mathbb {P}\left(A\right) = 1/3$, $\mathbb {P}\left(B\right) = 1/4$, $\mathbb {P}\left(AB\right) = 1/6$, find the following probabilities:

Given $\mathbb {P}\left(A\right) = 3/4$ and $\mathbb {P}\left(B\right) = 3/8$, show that:

For any two events $A$ and $B$ show that:

By induction show that for arbitrary events $A_{1}, A_{2}, \ldots , A_{n}$, the following inequality (due to Bonferoni) holds:

For any three events $A$, $B$, and $C$, show that (cf. formula (1.1), Poincaré's theorem for $n$ events $A_{1}, \ldots , A_{n}$: $\mathbb {P}\left(A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right) = S_{1} - S_{2} + \cdots + (-1)^{n-1}S_{n}$, where for each $k = 1, 2, \ldots , n$, $S_{k} = \sum_{i_{1} < i_{2} < \cdots < i_{k}} \mathbb {P}\left(A_{i_{1}}A_{i_{2}} \cdots A_{i_{k}}\right)$, the summation being over all combinations of the $n$ events taken $k$ at a time)

Application: The percentages of students who passed courses $A$, $B$, and $C$ are as follows: $A$: 50, $B$: 40, $C$: 30, $A$ and $B$: 35, $B$ and $C$: 20, and 15 passed all three courses. What is the percentage of students who succeeded in at least one of the three courses?

A box contains balls numbered $1, 2, \ldots , n$. A ball is drawn at random:

In terms of $\mathbb {P}\left(A\right)$, $\mathbb {P}\left(B\right)$, $\mathbb {P}\left(C\right)$, $\mathbb {P}\left(AB\right)$, $\mathbb {P}\left(AC\right)$, $\mathbb {P}\left(BC\right)$, and $\mathbb {P}\left(ABC\right)$ express, for $k = 0, 1, 2, 3$, the probabilities that:

1. exactly $k$ of the events $A$, $B$, and $C$, occur;

2. at least $k$ of the events $A$, $B$, $C$ occur.

By induction prove Poincaré's theorem, i.e., formula (1.1).

If $d(A,B) = \mathbb {P}\left(A \triangle B\right)$, show that $d$ has all the properties of a distance function; the symmetric difference $A \triangle B$ between two sets $A$ and $B$ is defined by $A \triangle B = AB' \cup A'B$.

Three winning tickets are drawn from an urn of 100 tickets. What is the probability of winning for a person who buys: (a) 4 tickets? (b) only one ticket?

A bakery makes 80 loaves of bread daily. Ten of them are underweight. An inspector weighs 5 loaves at random. What is the probability that an underweight loaf will be discovered?

Find the probability that among seven persons:

A group of $2N$ boys and $2N$ girls is randomly divided into two equal groups. What is the probability that each group has the same number of boys and girls?

The coefficients $a$, $b$, $c$ of the quadratic equation $ax^{2} + bx + c = 0$ are determined by throwing a die three times. Find the probabilities that: (a) the roots are real; (b) the roots are complex.

In the game of poker a ``hand of cards'' means 5 cards randomly selected (without replacement) from a deck of 52 cards. What is the probability that a hand of cards:

In bridge, a hand of cards'' consists of 13 cards drawn at random (without replacement) from a deck of 52 cards (the 52 cards are equally distributed to four players). Find the probabilities that a hand of cards'' will contain:

1. $v_{1}$ clubs, $v_{2}$ spades, and $v_{3}$ diamonds;

2. $v$ aces $(v = 0, 1, \ldots , 4)$;

3. $v_{1}$ aces and $v_{2}$ kings. In a bridge game find the probabilities that:

Six girls are to enter a dance with 10 boys to form a ring so that every girl is between two boys:

Five letters are selected at random one after another from the 26 letters of the English alphabet; (a) with replacement, (b) without replacement. Find, for each of the cases (a) and (b), the probabilities that the word formed: (i) contains an a'', (ii) consists of vowels, (iii) is the word woman''.

Prove that the number of ways in which $r$ indistinguishable balls can be distributed in $n$ cells (or, equivalently, the number of different solutions of the equation $x_{1} + x_{2} + \cdots + x_{n} = r$ where $x_{i} \geq 0$ $(i = 1, 2, \ldots , n)$ are integers), equals

Consider $r$ indistinguishable balls randomly distributed in $n$ cells (Bose--Einstein statistics). What is the probability that exactly $m$ cells remain empty?

From an ordinary deck of 52 cards, cards are drawn successively until an ace appears. What is the probability that the first ace will appear: (a) at the $n$th draw? (b) after the $n$th card?

In a classroom there are $v$ students.

$N$ letters are placed at random in $N$ envelopes. Show that the probability that each letter will be placed in a wrong envelope is $\displaystyle \sum_{k=2}^{N} (-1)^{k} \left(\frac{1}{k!}\right)$.

An urn contains $nr$ balls numbered $1, 2, \ldots , n$ in such a way that $r$ balls bear the same number $i$ for each $i = 1, 2, \ldots , n$. $N$ balls are drawn at random without replacement. Find the probability that,

(Continuation of Problem 49.) Balls are drawn until each of the numbers $1, 2, \ldots , n$ appears at least once. What is the probability that $m$ balls will be needed?

$N$ men run out of a men's club after a fire and each takes a coat and a hat. Prove that:

Suppose every packet of the detergent TIDE contains a coupon bearing one of the letters of the word TIDE. A customer who has all the letters of the word gets a free packet. All the letters have the same possibility of appearing in a packet. Find the probability that a housewife who buys 8 packets will get:

1. one free packet,

2. two free packets.

An urn contains $N_{1}$ white and $N_{2}$ black balls. When two balls are randomly drawn the probability that both be white is $1/2$.

If the events $A$, $B$ are independent, show that the pairs $(A, B')$, $(A', B)$, $(A', B')$ also consist of independent events.

If the events $A$, $B$, and $C$ are mutually independent, then the pairs $(A, BC)$, $(B, AC)$, $(C, AB)$ also consist of independent events.

Suppose that for the independent events $A$, $B$, and $C$ we have $\mathbb {P}\left(A\right) = a$, $\mathbb {P}\left(A \cup B \cup C\right) = 1 - b$, $\mathbb {P}\left(ABC\right) = 1 - c$, and $\mathbb {P}\left(A'B'C\right) = x$. Prove that the probability $x$ satisfies the equation

Hence conclude that

Moreover, show that

Let the events $A_{1}, A_{2}, \ldots , A_{n}$ be independent and $\mathbb {P}\left(A_{i}\right) = p$ ($i = 1, 2, \ldots , n$). What is the probability that:

A die is thrown as long as necessary for an ace or a 6 to turn up. Given that no ace turned up at the first two throws, what is the probability that at least three throws will be necessary?

A parent particle can be divided into 0, 1, or 2 particles with probabilities $1/4$, $1/2$, $1/4$, respectively. It disappears after splitting. Beginning with one particle, the progenitor, let us denote by $X_{i}$ the number of particles in the $i$ generation. Find,

An urn contains $n$ balls numbered $1, 2, \ldots , n$. We select at random $r$ balls,

Two athletic teams $A$ and $B$ play a series of independent games until one of them wins 4 games. The probability of each team winning in each game equals $1/2$. Find the probability that the series will end,

It is suspected that a patient has one of the diseases $A_{1}$, $A_{2}$, $A_{3}$. Suppose that the population percentages suffering from these illnesses are in the ratio $2:1:1$. The patient is given a test which turns out to be positive in $25\%$ of the cases of $A_{1}$, $50\%$ of $A_{2}$, and $90\%$ of $A_{3}$. Given that out of three tests taken by the patient two were positive, find the probability for each of the three illnesses.

The population of Nicosia (Cyprus) is $75\%$ Greek and $25\%$ Turkish. $20\%$ of the Greeks and $10\%$ of the Turks speak English. A visitor to the town meets someone who speaks English. What is the probability that he is a Greek? Interpret your answer in terms of the population of the town.

Two absent-minded room mates, mathematicians, forget their umbrellas in some way or another. $A$ always takes his umbrella when he goes out, while $B$ forgets to take his umbrella with probability $1/2$. Each of them forgets his umbrella at a shop with probability $1/4$. After visiting three shops they return home. Find the probability that:

Consider families of $n$ children and let $A$ be the event that a family has children of both sexes, and let $B$ be the event that there is at most one girl in the family. Show that the only value of $n$ for which the events $A$ and $B$ are independent is $n = 3$, assuming that each child has probability $1/2$ of being a boy.

At the college entrance examination each candidate is admitted or rejected according to whether he has passed or failed the test. Of the candidates who are really capable, $80\%$ pass the test; and of the incapable, $25\%$ pass the test. Given that $40\%$ of the candidates are really capable, find the proportion of capable college students.

$A$ and $B$ throw alternately a pair of dice in that order. $A$ wins if he scores 6 points before $B$ gets 7 points, in which case $B$ wins. If $A$ starts the game what is his probability of winning?

Three players $P_{1}$, $P_{2}$, and $P_{3}$ throw a die in that order and, by the rules of the game, the first one to obtain an ace will be the winner. Find their probabilities of winning.

$N$ players $A_{1}, A_{2}, \ldots , A_{N}$ throw a biased coin whose probability of heads equals $p$. $A_{1}$ starts (the game), $A_{2}$ plays second, etc. The first one to throw heads wins. Find the probability that $A_{k}$ ($k = 1, 2, \ldots , N$) will be the winner.

Urn $A$ contains $w_{1}$ white balls and $b_{1}$ black balls. Urn $B$ contains $w_{2}$ white balls and $b_{2}$ black balls. A ball is drawn from $A$ and is placed into $B$, and then a ball is transferred from $B$ to $A$. Finally, a ball is selected from $A$. What is the probability that the ball will be white?

Ten percent of a certain population suffer from a serious disease. A person suspected of the disease is given two independent tests. Each test makes a correct diagnosis $90\%$ of the time. Find the probability that the person really has the illness:

A player randomly chooses one of the coins $A$, $B$. Coin $A$ has probability of heads $3/4$, and coin $B$ has probability of heads $1/4$. He tosses the coin twice.

Urn $A$ contains 5 black balls and 6 white balls, and urn $B$ contains 8 black balls and 4 white balls. Two balls are transferred from $B$ to $A$ and then a ball is drawn from $A$.

The Pap test makes a correct diagnosis with probability $95\%$. Given that the test is positive for a lady, what is the probability that she really has the disease? How do you interpret this? Assume that one in every 2,000 women, on average, has the disease.

A secretary goes to work following one of three routes $A$, $B$, $C$. Her choice of route is independent of the weather. If it rains, the probabilities of arriving late, following $A$, $B$, $C$, are $0.06$, $0.15$, $0.12$, respectively. The corresponding probabilities, if it does not rain, are $0.05$, $0.10$, $0.15$.

At an art exhibition there are 12 paintings of which 10 are original. A visitor selects a painting at random and before he decides to buy, he asks the opinion of an expert about the authenticity of the painting. The expert is right in 9 out of 10 cases on average.

What is the conditional probability that a hand at poker (see Exercise 40) consists of spades, given that it consists of black cards?

Note: Of the 52 cards of an ordinary deck, the 13 spades and 13 clubs are black whereas the 13 hearts and 13 diamonds are red.

The probability $p_{k}$ that a family has $k$ children is given by $p_{0} = p_{1} = a$, $p_{k} = (1 - 2a)2^{-(k - 1)}$ ($k \geq 2$). It is known that a family has two boys. What is the probability that:

An amplifier may burn one or both of its fuses if one or both of its tubes are defective. Let the events:

The conditional probabilities of burning the fuses (given the state of the tubes, i.e., the $\mathbb {P}\left( B_{j} \mid A_{i} \right)$), appear in the following table:

Find the probabilities that:

An urn contains 7 white balls numbered $1, 2, \ldots , 7$ and 3 black balls numbered $8, 9, 10$. Five balls are randomly selected,

(Continuation of Problem 80.) An urn contains 7 white balls numbered $1, 2, \ldots , 7$ and 3 black balls, now numbered $1, 2$, and $3$ (instead of $8, 9, 10$ as in Problem 80). Five balls are randomly selected, (a) with replacement, (b) without replacement. As in Problem 80, give for each of the cases (a) and (b) the distribution:

1. of the number of white balls in the sample;

2. of the minimum number in the sample;

3. of the maximum number in the sample;

4. of the minimum number of balls needed for selecting a white ball.

A machine normally makes items of which $4\%$ are defective. Every hour the producer draws a sample of size 10 for inspection. If the sample contains no defective items he does not stop the machine. What is the probability that the machine will not be stopped when it has started producing items of which $10\%$ are defective.

One per thousand of a population is subject to certain kinds of accident each year. Given that an insurance company has insured 5,000 persons from the population, find the probability that at most 2 persons will incur this accident.

A certain airline company, having observed that $5\%$ of the persons making reservations on a flight do not show up for the flight, sells 100 seats on a plane that has 95 seats. What is the probability that there will be a seat available for every person who shows up for the flight?

Workers in a factory incur accidents at the rate of two accidents per week. Calculate the probability that there will be at most two accidents,

1. during 1 week,

2. during 2 weeks,

3. in each of 2 weeks.

Suppose that the suicide rate in a certain state is four suicides per one million inhabitants per month. Find the probability that in a certain town of population 500,000 there will be at most four suicides in a month. Would you find it surprising that during 1 year there were at least 2 months in which more than four suicides occurred?

A coin is tossed 30 times. For $n = 1, 2, \ldots , 20$, what is the conditional probability that exactly $10+n$ heads appear given that the first 10 tosses resulted in heads. Show that the conditional probability that $10+n$ tosses will result in heads, given that heads appeared at least ten times, equals

How many children should a family have so that with probability 0.95 it has at least a boy and at least a girl.

Show that the Poisson probabilities $p(k \mid \lambda )$ satisfy the recurrence relation

and hence determine the values of $k$ for which the terms $p(k \mid \lambda )$ reach their maximum (for given $\lambda$).

Verify that each of the following functions $f$ is a probability density function and sketch its graph.

The amount of bread (in hundreds of kilos) that a bakery sells in a day is a random variable with density

1. Find the value of $c$ which makes $f$ a probability density function.

2. What is the probability that the number of kilos of bread that will be sold in a day is, (a) more than 300 kilos? (b) between 150 and 450 kilos?

3. Denote by $A$ and $B$ the events in (a) and (b), respectively. Are $A$ and $B$ independent events?

Suppose that the duration in minutes of long-distance telephone conversations follows an exponential density function;

Find the probability that the duration of a conversation:

A number is randomly chosen from the interval $(0, 1)$. What is the probability that:

The height of men is normally distributed with mean $\mu = 167$ cm and standard deviation $\sigma = 3$ cm.

1. What is the percentage of the population of men that have height,

   1. greater than 167 cm,

   2. greater than 170 cm,

   3. between 161 cm and 173 cm?

2. In a random sample of four men what is the probability that:

   1. all will have height greater than 170 cm;

   2. two will have height smaller than the mean (and two bigger than the mean)?

A machine produces bolts the length of which (in centimeters) obeys a normal probability law with mean 5 and standard deviation $\sigma = 0.2$. A bolt is called defective if its length falls outside the interval $(4.8, 5.2)$.

Consider a shop at which customers arrive at random at a rate of twenty persons per hour. What is the probability that the time intervals between succesive arrivals will be:

If $X$ is a continuous random variable with cumulative distribution function $F$ and density function $f$, show that the random variable $Y = X^{2}$ is also continuous and express its cumulative distribution function and density in terms of $F$ and $f$.

(Continuation of Problem 97.) Find the density of $Y = X^{2}$ when $X$ has:

As in Exercises 97 and 98 with $Y = \left|X\right|$.

If the $\log X$ is normally distributed then $X$ is said to have a lognormal distribution. Find its density.

Show that the integral $\int_{-\infty }^{+\infty } \left|x - m\right| f(x)\, dx$ becomes minimum when $m$ is the median of the distribution with density $f$.

Let $f(x)$ denote the density function of the random variable $X$. Suppose that $X$ has a symmetric distribution about $a$, that is, $f(x + a) = f(a - x)$ for every $x$. Show that the mean $\mathbb {E}\left[X\right]$ equals $a$, provided it exists.

Show that $\mathbb {E}\left[XY\right]^{2} \leq \mathbb {E}\left[X^{2}\right] \mathbb {E}\left[Y^{2}\right]$ provided the second moments exist.

Show that $\left|\mathbb {E}\left[X\right]\right| \leq \mathbb {E}\left[\left|X\right|\right]$.

For every set $A$ of real numbers, we define the indicator function by

Show that

If $\mathbb {E}\left[X\right] = \mathbb {E}\left[X^{2}\right] = 0$, show that $\mathbb {P}\left(X = 0\right) = 1$.

Show that the mean $\mu$ of a random variable $X$ has the property