Miscellaneous Exercises

For any two events $A$, $B$ show that the following relationships are equivalent: $A \subset B$, $A' \supset B'$, $A \cup B = B$, and $A \cap B' = \emptyset$.

Show that $(A \triangle B)' = (AB) \cup (A'B')$ and $\mathbb {P}\left(A \triangle B\right) = \mathbb {P}\left(A\right) + \mathbb {P}\left(B\right) - 2\mathbb {P}\left(AB\right)$, where $\triangle$ is the symmetric difference, defined in Exercise 34.

Let $n$ be the number of elements in the sample space $\Omega$ and $N(E)$ the number of elements in the event $E$. Show that the data: $N = 1{,}000$, $N(A) = 525$, $N(B) = 312$, $N(C) = 470$, $N(AB) = 42$, $N(AC) = 147$, $N(BC) = 86$, and $N(ABC) = 25$ are inconsistent.

(Lewis Caroll, A Tangled Tale, 1881). In a fierce battle, at least 70 of the soldiers lost an eye, at least 74 lost an ear, at least 80 lost an arm, and at least 85 lost a leg. What is the smallest possible percentage of soldiers that each lost an eye, an ear, an arm, and a leg?

In a market research survey, from a sample of 1,000 smokers, 811 said that they prefer brand $A$'', 752 prefer $B$'', and 418 prefer $C$''. Furthermore, 570 said that they like both $A$'' and $B$'', 356 that they like both $A$'' and $C$'', 348 that they like both $B$'' and ``$C$'', and 297 that they like all three brands. How did the researcher discover that the results were inconsistent?

We throw two dice once. Let $A$: doubles, $B$: at least one die shows 5 or 6, $C$: the sum of the outcomes of the two dice is less than 6.

Three athletes are participating in the Marathon. Their chances of finishing under 2 hours are $1/3$, $1/5$, and $1/12$, respectively. Their times for the duration of the course are independent. What is the probability that at least one finishes in less than 2 hours?

A horse has probability $p$ of jumping over a specific obstacle. Given that in 5 trials it succeeded 3 times, what is the conditional probability that it had succeeded in the first trial?

In a train car 3 seats are facing the front and 3 are facing the rear. Two women and three men enter the car and sit at random. What is the probability:

On the basis of the following data, examine the relationship between the father having brown eyes (event $A$) and the son having brown eyes (event $B$). The percentages observed were: $AB$: 5, $AB'$: 7.9, $A'B$: 8.9, and $A'B'$: 78.2.

We throw three dice. What is the probability that at least one 6 appears given that the three outcomes were completely different. Generalize the result for the case of a regular polyhedral die.

Ninety percent of a population have a television set (event $A$), while 80 have a car (event $B$). What is the minimum percentage of the population with a car that also have a television?

Let $\mathbb {P}\left(A\right) = \alpha$ and $\mathbb {P}\left(B\right) = 1 - \varepsilon$ where $\varepsilon$ is a small number. Give the upper and lower bound of $\mathbb {P}\left( A \mid B \right)$.

Let $X_{1}, X_{2}$ be independent random Bernoulli variables with the same parameter $p$ and let $Y = 0$ if $X_{1} + X_{2} =$ even and $Y = 1$ if $X_{1} + X_{2} =$ odd.

We throw three dice twice. What is the probability of having the same result the second time when the dice are, (a) not distinguishable? (b) distinguishable?

At a party, $A$ is introduced to 6 women and 4 men. What is the probability that he identifies all the couples correctly if he knows that there are, (a) 4 couples, (b) 3 couples?

A young person driving a motorcycle on a highway with 4 lanes changes lane every 15 seconds. What is the probability $p$ of finding himself in the lane in which he started 1 minute later?

In city $A$ it rains half of the days of the year and the weather forcast is correct with probability $2/3$. Mr. Sugar worrying about getting wet always takes his umbrella when the forcast calls for rain and with probability $1/3$ otherwise. Find the probability that:

The telegram signals $(\cdot )$ and $(-)$ are sent with proportion $3\! :\! 4$. Due to faults in transmission a dot $(\cdot )$ becomes $(-)$ with probability $1/3$. If a signal has arrived as $(\cdot )$ what is the probability that it was transmitted correctly?

A die is thrown 10 times and let the maximum outcome be $M$ and the minimum $m$. Find the $\mathbb {P}\left(m = 2, M = 5\right)$.

An urn contains $b$ black balls and $w$ white balls. $n$ balls are drawn from the urn at random, their color unnoticed, then $m$ additional balls are drawn. Show that the probability that there are $k$ black balls among the $m$ balls is the same as if the $m$ balls were chosen from the beginning (i.e., before choosing the $n$ balls).

In other words, in sampling from a finite population the final (hypergeometric) probabilities are not affected by the content of an initial sample when we do not know that content. Obviously, if we know the composition of the first sample (i.e., the colors of the $n$ balls), then, in general, the probabilities are affected.

After an earthquake the books of a large library fall on the floor and the servant, who does not know how to read, reshelves them in any order.

John Smith in 1693 posed the following problem: Is the probability for each of the three players to win the same if: the first must obtain at least one ace throwing 6 dice, the second at least two aces throwing 12 dice, and the third at least three aces throwing 18 dice? Newton and Tollet found that the first player has a better chance than the second one, and the second player has a better chance than the third one.

A homogeneous and regular die with $v$ faces is thrown $n$ times. Let $p(n, k)$ be the probability that a given face appears less than $k$ times. Show that:

The Massachusetts State Lottery issues 1 million tickets. The first winning ticket wins $50,000, the next 9 winning tickets win $2,500 each, the next 90 tickets win $250 each, and 900 tickets win $25 each. What is the expected gain of a person buying, (a) one ticket? (b) ten tickets?

(Continuation of Problem I-25.) What is the answer to (a) and (b) if:

1. the state issues 2 million tickets?

2. the winning tickets win twice as much?

In a survey on the housing problem, 2 apartments are chosen from each of 4 buildings (having both owner-occupied as well as rented apartments). If we know that:

What is the expected number of owner-occupied apartments in the above sample?

Out of 365 people:

The duration $T$ of a telephone call satisfies the following equation:

where $0 \leq p \leq 1$, $\lambda > 0$, $\mu > 0$. Find $\mathbb {E}\left[T\right]$ and $\operatorname {Var}\left[T\right]$.

Two gamblers $A$ and $B$ play a series of independent games in each of which $A$ has probability $p$ of winning and $B$ has probability $q = 1 - p$ of winning. Suppose that at a given moment the series of games is interrupted and $A$ needs $m$ additional points and $B$ needs $n$ additional points to be the winner. The bet should be split between $A$ and $B$ in the ratio $\mathbb {P}\left(A\right):\mathbb {P}\left(B\right)$, where $\mathbb {P}\left(A\right) = 1 - \mathbb {P}\left(B\right)$ is the probability that $A$ eventually wins. Show that

Note: The above solution was first obtained by Montmort (1675--1719). Explain the idea behind each term in the expressions for $P_{A}$. Try to show the equality between those two expressions for $P_{A}$ using the properties of binomial coefficients (see Chapter 1).

An employee leaves between 7:30 a.m. and 8:00 a.m. to go to his office, and he needs 20--30 minutes to get to the railway station. The departure time and the duration of travel are distributed independently and uniformly in the corresponding intervals. He can take either of the following two trains: the first leaves at 8:05 a.m. and takes 35 minutes; and the second leaves at 8:25 a.m. and takes 30 minutes.

Show that if $X$ and $Y$ are independent, then (cf. Exercise 328)

Conclude that if $X_{1}, \ldots , X_{n}$ are completely independent with means zero, then

We toss a fair coin until two consecutive heads or tails appear. What is the probability that an even number of tosses will be required?

$s$ distinguishable balls are placed randomly into $n$ different cells. Find the probability that exactly $r$ cells will be occupied.

A large board is divided into equal squares of side $a$. A coin of diameter $2r < a$ is thrown randomly onto the board. Find the probability that the coin overlaps:

1. with exactly one square;

2. with exactly two squares.

A square of side $a$ is inscribed into a circle. Find the probability that:

1. any randomly selected point of the circle will be an interior point of the square.

2. of eight randomly selected points in the circle, three fall inside the square, two in one of the four remaining parts of the circle, and one in each of the other three parts of the circle.

The distance between two successive cars on a highway follows the exponential distribution with mean $20m$. Find the probability that there are 90 to 110 cars on a 2-km stretch of the highway.

The number of cars crossing a toll bridge follows the Poisson distribution with an average of 30 cars per minute. Find the probability that in 200 seconds no more than 100 cars will pass through.

(Continuation of Problem I-38.) Establish the relation

between the Erlang distribution function and the Poisson distribution function.

(Continuation of Problem I-39.) Show that

Define $\Phi (a; b) = \Phi (b) - \Phi (a)$ for $0 < a < b$ with $\Phi$ the distribution function of $\mathscr {N}\left(0, 1\right)$. Show that $\Phi (0; 1) > \Phi (1; 2)$. Generalize this result for any equal intervals.

Using the fact that if $X$ follows the normal distribution $\mathscr {N}\left(0, 1\right)$, then

conclude that

The probability of hitting a target with one shot is $p_{1}$ and the probability of destroying the target with $k$ ($k \geq 1$) shots is $1 - p_{2}^{k}$. Find the probability of destroying the target after $n$ shots.

Let $X$ follow a Poisson distribution with parameter $\lambda$. We select $N$ random points from the interval $[0, 1]$. Let $X_{i}$ be the number of points in the interval $[(i-1)/n, i/n]$ ($i = 1, \ldots , n$). Show that the random variables $X_{i}$ are independent and identically distributed.

If $X$ and $Y$ are independent with $\mathbb {P}\left(X = 0\right) = \mathbb {P}\left(X = 1\right) = 1/2$ and $\mathbb {P}\left(Y \leq y\right) = y$ ($0 \leq y \leq 1$), find the distribution of the following random variables $W$, $Z$ where:

1. $W = X + Y$;

2. $Z = XY$.

Let $X$ be a uniform random variable on the interval $(-a, a)$ and $Y$ a random variable independent of $X$, with distribution function $F(y)$. Show that the distribution function of $G = X + Y$ is

Let $X$ be uniformly distributed on the interval $(0, 1)$. Find the probability density function of:

1. $Y = aX + b$; $a, b$ constants ($a > 0$);

2. $Z = 1/X$;

3. $W = g(X)$, where $g$ is a continuous and monotone function of $x$ in $(0, 1)$.

Each of three players $A$, $B$, and $C$ has probability $1/3$ of hitting a target better than the other two. At each round of the game each shoots at the target once. The winner is the first player to win six rounds. They bet 10 ducats each. At a certain point, however, when $A$ has 4 victories, $B$ has 3 victories, and $C$ has 2 victories, they have to terminate the game. How should they split their bet? (cf. Problem 30).

$A$ and $B$ play a game in which each of them throws two ordinary dice. $A$ wins if he scores 6 and $B$ wins if he scores 7. First, $A$ throws the dice once, then $B$ throws the dice twice, and they continue alternating after each throw until one of them wins. What is the probability of winning for each of them?

Note: Huyghens and Spinoza gave the same answer:

(cf. Exercise 67).

A deck of forty cards consists of 10 red, 10 black, 10 blue, and 10 green cards. Four cards are drawn at random and without replacement. What is the probability that all the colors will show up?

(Huyghens gave the answer $1{,}000/9{,}129 = 0.109$; is it correct?)

An urn contains 8 black balls and 4 white balls. Each of three players $A$, $B$, and $C$ (and in that order) draws a ball without replacement and the first one to draw a white ball wins. What is the probability of winning for each of them?

Three players $A$, $B$, and $C$ play the following game. They play three rounds for each of which, each of them has probability $1/3$ of winning. $A$ is the winner of the game if he wins a round before $B$ or $C$ wins two rounds; $B$ is the winner if he wins two rounds before player $A$ wins one round, or $C$ wins two rounds; $C$ wins like $B$. What is the probability of winning for each of them?

(Montmort gave $\mathbb {P}\left(A\right) = 15/27$, $\mathbb {P}\left(B\right) = \mathbb {P}\left(C\right) = 5/27$).

A container has $2N$ cards, numbered in such a way that each of the numbers $1, 2, \ldots , N$ appears twice. $2N - n$ cards are drawn at random. What is the expected number of pairs of cards remaining in the container?

(Bernoulli found $n(n-1)/(4N-2)$).

$N$ lottery tickets are numbered 1 to $N$. $n$ tickets are selected at random. What is the probability that $r$ consecutive numbers are selected ($r = 1, 2, \ldots , n$)?

The probability of event $A$ is $p$. What is the probability that:

In a lottery with $N$ tickets, $n$ numbers are chosen as winners. What is the probability $p$ that in $k$ lottery games all the $N$ numbers will appear?

(Laplace for $N = 90$, $n = 5$, and $k = 85$ gave $p = 1/2$).