Question 1

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?

Accepted Answer

The answer of An American household is chosen at random...

Question 2

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?

Accepted Answer

The answer of Consider the following game. You pay me...

Question 3

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?

Accepted Answer

The answer of Here is an assignment of probabilities to...

Question 4

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?

Accepted Answer

The expected net value of a raffle ticket is calculated as the probability of winning multiplied by the value of the prize minus the cost of the ticket. The probability of winning is 1/150, and the value of the prize is $8,000. Therefore, the expected net value of a raffle ticket is:

(1/150) x $8,000 - $100 = -$46.67

This means that on average, a person buying a raffle ticket can expect to lose $46.67. Therefore, the best choice is not to buy a raffle ticket.

Question 5

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?

Accepted Answer

The answer of In many popular board games, a player...

Question 6

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").

Accepted Answer

The expected net gain is calculated as (0.10 * $4) + (0.90 * -$1) = $0.40 - $0.90 = -$0.50.

Question 7

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?

Accepted Answer

The answer of In many popular board games, a player...

Question 8

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?

Accepted Answer

The expected number of right answers is the sum of the products of each possible number of correct answers and its corresponding probability. Thus, E(X) = 0.002(0) + 0.018(1) + 0.087(2) + 0.235(3) + 0.336(4) + 0.250(5) + 0.084(6) + 0.008(7) ≈ 2.493. Therefore, the best choice is B.

Question 9

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?

Accepted Answer

In order to determine the expected net value per ticket, we need to calculate the probability of winning and the amount of money gained or lost. For the first raffle, the probability of winning is 1/150 and the net gain is $7,900 ($8,000 prize minus the $100 cost of the ticket). Therefore, the expected net value per ticket is (1/150) x $7,900 = $52.67. For the second raffle, the probability of winning is 1/8000 and the net gain is $349,900 ($350,000 prize minus the $100 cost of the ticket). Therefore, the expected net value per ticket is (1/8000) x $349,900 = $43.74. Since the expected net value per ticket is higher for the first raffle, it is the better choice.

Question 10

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.

Accepted Answer

The expected value for a ticket buyer is calculated by subtracting the cost of the ticket from the probability-weighted average of all possible outcomes. The probability of winning is 1/150, and the prize is $8,000, so the expected winnings are $8,000 * (1/150) = $53.33. Subtracting the cost of the ticket ($100) from the expected winnings ($53.33) results in an expected value of -$46.67 for each ticket purchased.

Quiz 20: The House Edge: Expected Values

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?

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 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?

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?

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 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.