Deck 20: The House Edge: Expected Values

Question 1

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) the probability of winning a bet.
B) the bias of a bet.
C) a random number.
D) the expected value of a bet.

Answer 1

the expected value of a bet.

Question 2

An employee's office window overlooks a busy four-way stop, and the employee decides to track, per minute, the number of drivers who fail to stop. From data taken over many randomly selected times, the employee gets this probability model:

In Question 14, the employee found the expected number of drivers failing to stop, per minute. The law of large numbers says:

A) observe whether each of many drivers fails to stop. The proportion who fail to stop will be close to the number calculated in the previous question.
B) if five observations of failure to stop are made in a row, the next several observations are likely to have four or five drivers failing to stop because the average must stay close to the expected value.
C) the expected value is correct only in a randomized comparative experiment.
D) probabilities can be found by simulation.
E) many observations should be made and the number of drivers failing to stop should be recorded in each case. The average will be close to the number calculated in the previous question.

Answer 2

observe whether each of many drivers fails to stop. The proportion who fail to stop will be close to the number calculated in the previous question.

Question 3

A friend rolls a cheap six-sided die many times. He reports that the probabilities of the possible outcomes are about as follows:

Using the probabilities above, what is the expected outcome when rolling this die?

A) 3
B) 3.3
C) 3.5
D) 4
E) 4.5

Answer 3

3.3

Question 4

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) about -$1, because you will lose most of the time
B) about $9, because you win $10 but lose only $1
C) about -$0.15, that is, on the average you lose about 15 cents per play
D) about $0.77, that is, on the average you win about 77 cents per play
E) about $0, because the random draw gives you a fair bet

Question 5

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.

A) He's right, because the winnings have to average out.
B) He's wrong, because successive deals are independent of each other.
C) He's right, because successive deals are independent of each other.
D) He's wrong, because his expected winnings are $0 and he's below that now.

Question 6

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) Wager 1, because it has a higher expected value
B) Wager 2, because it has a higher expected value
C) Wager 1, because it has a higher probability of winning
D) Wager 2, because it has a higher probability of winning
E) Both wagers are equally favorable.

Question 7

In government data, a family consists of two or more persons who live together and are related by blood or marriage. Choose an American family at random and count the number of people it contains. Here is the assignment of probabilities for your outcome:

Using the probabilities above, what is the expected size of the family you draw?

A) 2 people
B) 3 people
C) 3.14 people
D) 3.5 people
E) 4.5 people

Question 8

Kevin thinks he can use ESP to predict the outcome of rolling a fair die. You agree to pay him $3 if he can correctly predict the results of the next roll. Kevin has to pay you $1 if he is wrong. If Kevin doesn't have any psychic powers, which of the following is closest to the expected value of your net winnings on this bet?

A) -$0.67
B) -$0.33
C) $0
D) $0.33
E) $0.67

Question 9

Suppose you know the percentage of foul shots a basketball player makes during the season. You want to estimate the expected number of shots made in 10 shots. You simulate 10 shots 25 times and get the following numbers of shots made:

Your estimate is:

A) 5.5
B) 6
C) 6.4
D) 6.7
E) 7

Question 10

A home game for a certain National League Central Division team is chosen at random. Here is the probability distribution for the number of hits the leadoff batter has had:

What is the expected value of the number of hits the leadoff batter will have in a randomly chosen game?

A) 1.00
B) 1.02
C) 1.42
D) 1.97
E) 2.5

Question 11

You play a game with two possible outcomes. Outcome A has probability 0.4 and outcome B has probability 0.6. When B occurs you win $2.00; otherwise, you lose $1.00. What is your expected value for this game?

A) $2.00
B) -$0.10
C) $0.20
D) -$0.80
E) $0.80

Question 12

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

A) absence of bias.
B) absence of confounding.
C) law of large numbers.
D) independence.
E) expected value.

Question 13

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.
What is the expected number of right answers Paul will get if the test has 20 questions?

A) 5
B) 4
C) 2
D) 1/4
E) Can't tell from the information given.

Question 14

Which of the following statements about a table of random digits is true?

A) Statement I: If each line contains 40 digits, there will be exactly 4 zeros in every line.
B) Statement II: The probability that there are exactly 4 zeros in a line of 40 digits is exactly 0.5.
C) Statement III: The expected number of zeros in a line of 40 digits is 4.
D) Statement IV: There can never be 4 zeros in a row because that pattern isn't random.
E) Both statements III and IV are true.

Question 15

A grocery chain runs a prize game by giving each customer a ticket that may win a prize when a box is scratched. Printed on the ticket are the following probabilities for a customer who shops once a week:

If several thousand customers play the grocery store game, you expect that the mean amount they win will be close to

A) the probability of winning something.
B) the median amount they win.
C) the expected value of a customer's winnings.
D) $100.

Question 16

A game involving a pair of dice pays you $4 with probability 16/36, costs you $2 with probability 14/36, and costs you $6 with probability 6/36.
What is your expected net result, in dollars, per play?

A) -$8
B) -$1.78
C) $0
D) $1.78
E) $8

Question 17

A grocery chain runs a prize game by giving each customer a ticket that may win a prize when a box is scratched. Printed on the ticket are the following probabilities for a customer who shops once a week:

What is the expected value of a customer's winnings in this game?

A) $22.00
B) $10.00
C) $370.00
D) $0.31
E) $100

Question 18

Choose an American household at random and ask how many computers that household owns. Here are the probabilities as of 2009:

What is the expected number of computers owned by a randomly chosen household?

A) 1.412
B) 1.307
C) 2.214
D) 2
E) 2.5

Question 19

A game involving a pair of dice pays you $4 with probability 16/36, costs you $2 with probability 14/36, and costs you $6 with probability 6/36.
If you play this game many times, in the long run how will your actual average gain per play compare with your answer to the previous question?

A) smaller
B) bigger
C) about the same
D) The question can't be answered.

Question 20

An employee's office window overlooks a busy four-way stop, and the employee decides to track, per minute, the number of drivers who fail to stop. From data taken over many randomly selected times, the employee gets this probability model:

The expected number of drivers failing to stop, per minute, is:

A) 0.5.
B) 2.2.
C)2)
D)3)
E) varies from minute to minute.

Question 21

An American household is chosen at random and asked how many tablet computers it owns. Here are the probabilities as determined by a recent survey:

What is the expected number of tablet computers per household?

A) 1
B) 1.44
C) 1.5
D) 1.54
E) 2

Question 22

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) $2
B) $4
C) $1
D) $0.75
E) $0.50

Question 23

Here is an assignment of probabilities to the face that comes up when rolling a die once:

What is the approximate expected outcome when rolling this die?

A) 3
B) 3.29
C) 3.5
D) 4
E) 4.29

Question 24

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) -$46.00
B) -$46.67
C) $100
D) $7,900
E) $8,000

Question 25

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) 6
B) 7
C) 6.5
D) 6/36
E) 42/36

Question 26

Consider the following lottery game. For this game, tickets cost $1, and there is a 10% chance of winning $5 (a net gain of $4). What is a player's expected net gain for each ticket that is bought for this game? (Note that a loss results in a net gain of "-$1").

A) $5
B) $4
C) $0.50
D) -$0.50
E) -$1

Question 27

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: Suppose you pay $5 to roll the two dice, and you'll win a dollar amount equal to the sum of the two dice. What is your expected profit for this game?

A) $2
B) $5
C) $4
D) $6.50
E) $7

Question 28

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) 2.282
B) 2.493
C) 3.5
D) 5
E) Can't tell from the information given.

Question 29

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.
A second hospital is also conducting a benefit raffle to raise funds. They have 8000 tickets that also cost $100 each, and one ticket will be drawn at random also for a $350,000 prize-a new custom home. Which raffle has the best value for you per ticket?

A) Neither one, the expected net value of a ticket is the same for both raffles.
B) The first raffle because there are fewer tickets from which to choose a winner.
C) The second raffle because the prize is bigger.
D) The first raffle because the expected net value per ticket is higher.
E) The second raffle because the expected net value per ticket is higher.

Question 30

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.

A) $8,000
B) $7,000
C) $100
D) $46
E) $46.67