A card is drawn at random from a deck consisting of cards numbered 2 through 10. A player wins 1 dollar if the number on the card is odd and loses 1 dollar if the number if even. What is the expected value of his winnings?

A card is drawn at random from a deck of playing cards. If it is red, the player wins 1 dollar; if it is black, the player loses 2 dollars. Find the expected value of the game.

In a class there are 20 students: 3 are 5' 6”, 5 are 5'8”, 4 are 5'10”, 4 are 6', and 4 are 6' 2”. A student is chosen at random. What is the student's expected height?

In Las Vegas the roulette wheel has a 0 and a 00 and then the numbers 1 to 36 marked on equal slots; the wheel is spun and a ball stops randomly in one slot. When a player bets 1 dollar on a number, he receives 36 dollars if the ball stops on this number, for a net gain of 35 dollars; otherwise, he loses his dollar bet. Find the expected value for his winnings.

In a second version of roulette in Las Vegas, a player bets on red or black. Half of the numbers from 1 to 36 are red, and half are black. If a player bets a dollar on black, and if the ball stops on a black number, he gets his dollar back and another dollar. If the ball stops on a red number or on 0 or 00 he loses his dollar. Find the expected winnings for this bet.

A die is rolled twice. Let [math]X[/math] denote the sum of the two numbers that turn up, and [math]Y[/math] the difference of the numbers (specifically, the number on the first roll minus the number on the second). Show that [math]E(XY) = E(X)E(Y)[/math]. Are [math]X[/math] and [math]Y[/math] independent?

Show that, if [math]X[/math] and [math]Y[/math] are random variables taking on only two values each, and if [math]E(XY) = E(X)E(Y)[/math], then [math]X[/math] and [math]Y[/math] are independent.

A royal family has children until it has a boy or until it has three children, whichever comes first. Assume that each child is a boy with probability 1/2. Find the expected number of boys in this royal family and the expected number of girls.

If the first roll in a game of craps is neither a natural nor craps, the player can make an additional bet, equal to his original one, that he will make his point before a seven turns up. If his point is four or ten he is paid off at [math]2 : 1[/math] odds; if it is a five or nine he is paid off at odds [math]3 : 2[/math]; and if it is a six or eight he is paid off at odds [math]6 : 5[/math]. Find the player's expected winnings if he makes this additional bet when he has the opportunity.

In Example assume that Mr. Ace decides to buy the stock and hold it until it goes up 1 dollar and then sell and not buy again. Modify the program StockSystem to find the distribution of his profit under this system after a twenty-day period. Find the expected profit and the probability that he comes out ahead.