Exam 20: The House Edge: Expected Values

A children's cancer center sells 150 raffle tickets to raise money. Tickets cost $100 each. One ticket will be drawn at random for the $8,000 prize: a new men's Rolex watch.

Consider the following game. You pay me an entry fee of x dollars, then I roll a fair die. If the die shows a number less than 3, I pay you nothing; if the die shows a 3 or 4, I give you back your entry fee of x dollars; if the die shows a 5, I will pay you $1; and if the die shows a 6, I pay you $3. What value of x makes the game fair (in terms of expected value) for both of us?

A multiple choice exam offers four choices for each question. Paul just guesses the answers, so he has probability 1/4 of getting any one answer right. Paul's guess on any one question gives no information about his guess on any other question. The statistical term for this is

In many popular board games, a player rolls two dice and moves the number of spaces equal to the sum shown on the dice. Here is the assignment of probabilities to the sum of the numbers on the up faces when two dice are rolled: What is the expected sum when rolling two dice?

A standard deck of cards contains 52 cards, of which 4 are aces. You are offered the following wager: draw one card at random from the deck. You win $10 if the card drawn is an ace. Otherwise, you lose $1. If you make this wager very many times, what will be the mean outcome?

A children's cancer center sells 150 raffle tickets to raise money. Tickets cost $100 each. One ticket will be drawn at random for the $8,000 prize: a new men's Rolex watch. What is the expected net value of a raffle ticket?

A gambler who keeps placing $1 bets on roulette will, after a very large number of bets, find that his average winnings per bet are close to $0.947. (The house keeps the other $0.053 per bet.) The statistical term for the number $0.947 is

A standard deck of 52 cards contains 13 hearts. Here is another wager: draw one card at random from the deck. If the card drawn is a heart, you win $2. Otherwise, you lose $1. Compare this wager (call it Wager 2) with that of the previous question (call it Wager 1). Which one should you prefer?

A multiple-choice exam offers five choices for each question. Jason just guesses the answers, so he has probability 1/5 of getting any one answer right. One of your math major friends tells you that the assignment of probabilities to the number of questions Jason gets right out of 10 is (rounded to three decimal places): What is the expected number of right answers Jason will get if the test has 10 questions?

A poker player is dealt poor hands for several hours. He decides to bet heavily on the last hand of the evening on the grounds that after many bad hands he is due for a winner.