A newly married couple decides to continue having children

There is a story about Charles Dickens (18121870), the English novelist and one of the most popular writers in the history of literature. It is known that Dickens was interested in practical applications of mathematics. On the final day in March during a year in the second half of the nineteenth century, he was scheduled to leave London by train and travel about an hour to visit a very good friend. However, Mr. Dickens was aware of the fact that in England there were, on the average, two serious train accidents each month. Knowing that there had been only one serious accident so far during the month of March, Dickens thought that the probability of a serious train accident on the last day of March would be very high. Thus he called his friend and postponed his visit until the next day. He boarded the train on April 1, feeling much safer and believing that he had used his knowledge of mathematics correctly by leaving the next day. He did arrive safely! Is there a fallacy in Dickens, argument? Explain.

In a certain part of downtown Baltimore parking lots charge $7 per day. A car that is illegally parked on the street will be fined $25 if caught, and the chance of being caught is 60%. If money is the only concern of a commuter who must park in this location every day, should he park at a lot or park illegally?

In a lottery every week, 2,000,000 tickets are sold for $1 apiece. If 4000 of these tickets pay off $30 each, 500 pay off $800 each, one ticket pays off $1,200,000, and no ticket pays off more than one prize, what is the expected value of the winning amount for a player with a single ticket?

In a lottery, a player pays $1 and selects four distinct numbers from 0 to 9. Then, from an urn containing 10 identical balls numbered from 0 to 9, four balls are drawn at random and without replacement. If the numbers of three or all four of these balls matches the players numbers, he wins $5 and $10, respectively. Otherwise, he loses. On the average, how much money does the player gain per game? (Gain = win loss.)

An urn contains five balls, two of which are marked $1, two $5, and one $15. A game is played by paying $10 for winning the sum of the amounts marked on two balls selected randomly from the urn. Is this a fair game?

A box contains 20 fuses, of which five are defective. What is the expected number of defective items among three fuses selected randomly?

The demand for a certain weekly magazine at a newsstand is a random variable with probability mass function p(i) = (10i)/18, i = 4, 5, 6, 7. If the magazine sells for $a and costs $2a/3 to the owner, and the unsold magazines cannot be returned, how many magazines should be ordered every week to maximize the profit in the long run?

It is well known that / x=1 1/x2 = 2/6. (a) Show that p(x) = 6/(x)2, x = 1, 2, 3,... is the probability mass function of a random variable X. (b) Prove that E(X) does not exist.

(a) Show that p(x) = ! |x| + 1 "2 /27, x = 2, 1, 0, 1, 2, is the probability mass function of a random variable X. (b) Calculate E(X), E ! |X| " , and E(2X2 5X + 7).

A box contains 10 disks of radii 1, 2, . . . , 10, respectively. What is the expected value of the circumference of a disk selected at random from this box?

The distribution function of a random variable X is given by F (x) = 0 if x < 3 3/8 if 3 x < 0 1/2 if 0 x < 3 3/4 if 3 x < 4 1 if x 4. Calculate E(X), E ! X2 2|X| " , and E ! X|X| " .

If X is a random number selected from the first 10 positive integers, what is E 4 X(11 X)5 ?

Let X be the number of different birthdays among four persons selected randomly. Find E(X).

A newly married couple decides to continue having children until they have one of each sex. If the events of having a boy and a girl are independent and equiprobable, how many children should this couple expect? Hint: Note that / i=1 iri = r/(1 r)2, |r| < 1.

Suppose that there exist N families on the earth and that the maximum number of children a family has is c. For j = 0, 1, 2, . . . , c, let j be the fraction of families with j children !/c j=0 j = 1 " . A child is selected at random from the set of all children in the world. Let this child be the Kth born of his or her family; then K is a random variable. Find E(K).

Suppose that there exist N families on the earth and that the maximum number of children a family has is c. For j = 0, 1, 2, . . . , c, let j be the fraction of families with j children !/c j=0 j = 1 " . A child is selected at random from the set of all children in the world. Let this child be the Kth born of his or her family; then K is a random variable. Find E(K).

Suppose that n random integers are selected from {1, 2, . . . , N} with replacement. What is the expected value of the largest number selected? Show that for large N the answer is approximately nN/(n + 1).

(a) Show that p(n) = 1 n(n + 1) , n 1, is a probability mass function. (b) Let X be a random variable with probability mass function p given in part (a); find E(X).

To an engineering class containing 2n 3 male and three female students, there are n work stations available. To assign each workstation to two students, the professor forms n teams one at a time, each consisting of two randomly selected students. In this process, let X be the number of students selected until a team of a male and a female is formed. Find the expected value of X.