Question
Question 1 (1 point)
As posted on the Maryland Lottery's website for its Pick 3 game, the chance of winning with a Front Pair bet is 0.01. A Front Pair bet is successful if the front pair of numbers you select match the Pick 3 number's first 2 digits. For example, a bet of 12__ would be a winner if the Pick 3 number is 120, 121, 122, and so on of any 3 digit number beginning with 12. In other words, 10 of the 1,000 possible Pick 3 numbers (or 1%) would be winners, and thus, the probability of winning is 0.01 or 1%. A successful bet of $0.50 pays out $25 to the player.
If a person plays the lottery one time, compute the person's expected payout.
Question 1 options:
-$0.50
$0.50
-$0.25
$0.25
Question 2 (1 point)
You are invited to play a simple game of chance with a friend. The rules are:
To play you must pay him $2.
If you pick a spade from a shuffled standard deck of cards, you win $9.
What is the expected payoff you could win (or lose) per game?
Question 2 options:
You would lose $2.00 on this game.
You would win $0.25 on this game.
You would win $2.25 on this game.
You would lose $0.25 on this game.
Question 3 (1 point)
As posted on the Maryland Lottery's website for its Pick 3 game, the chance of winning with a Front Pair bet is 0.01. A Front Pair bet is successful if the front pair of numbers you select match the Pick 3 number's first 2 digits. For example, a bet of 12__ would be a winner if the Pick 3 number is 120, 121, 122, and so on of any 3 digit number beginning with 12. In other words, 10 of the 1,000 possible Pick 3 numbers (or 1%) would be winners, and thus, the probability of winning is 0.01 or 1%. A successful bet of $0.50 pays out $25 to the player.
Assume that for a given time period, 100,000 bets like the one described above were placed. How much money should the Maryland Lottery Agency expect to earn on an average from 100,000 bets?
Question 3 options:
$25,000
$50,000
$100,000
-$25,000
Question 4 (1 point)
A $20 bill, two $10 bills, three $5 bills, and four $1 bills are placed in a bag. If a bill is chosen at random, what is the expected value for the amount chosen? If you managed this game, how much would you charge people to play? Explain your reasoning.
Question 4 options:
You would want to charge people at least $20 to play.
You would want to charge people more than $5.90 to play.
You would want to charge people less than $5.90 to play.
You would want to charge people exactly $5.90 to play.
Question 5 (1 point)
As posted on the Maryland Lottery's website for its Pick 3 game, the chance of winning with a Front Pair bet is 0.01. A Front Pair bet is successful if the front pair of numbers you select match the Pick 3 number's first 2 digits. For example, a bet of 12__ would be a winner if the Pick 3 number is 120, 121, 122, and so on of any 3 digit number beginning with 12. In other words, 10 of the 1,000 possible Pick 3 numbers (or 1%) would be winners, and thus, the probability of winning is 0.01 or 1%. A successful bet of $0.50 pays out $25 to the player.
On average, how much does the Maryland lottery make on one bet?
Question 5 options:
-$0.25
$0.25
-$0.50
$0.50
As posted on the Maryland Lottery's website for its Pick 3 game, the chance of winning with a Front Pair bet is 0.01. A Front Pair bet is successful if the front pair of numbers you select match the Pick 3 number's first 2 digits. For example, a bet of 12__ would be a winner if the Pick 3 number is 120, 121, 122, and so on of any 3 digit number beginning with 12. In other words, 10 of the 1,000 possible Pick 3 numbers (or 1%) would be winners, and thus, the probability of winning is 0.01 or 1%. A successful bet of $0.50 pays out $25 to the player.
If a person plays the lottery one time, compute the person's expected payout.
Question 1 options:
-$0.50
$0.50
-$0.25
$0.25
Question 2 (1 point)
You are invited to play a simple game of chance with a friend. The rules are:
To play you must pay him $2.
If you pick a spade from a shuffled standard deck of cards, you win $9.
What is the expected payoff you could win (or lose) per game?
Question 2 options:
You would lose $2.00 on this game.
You would win $0.25 on this game.
You would win $2.25 on this game.
You would lose $0.25 on this game.
Question 3 (1 point)
As posted on the Maryland Lottery's website for its Pick 3 game, the chance of winning with a Front Pair bet is 0.01. A Front Pair bet is successful if the front pair of numbers you select match the Pick 3 number's first 2 digits. For example, a bet of 12__ would be a winner if the Pick 3 number is 120, 121, 122, and so on of any 3 digit number beginning with 12. In other words, 10 of the 1,000 possible Pick 3 numbers (or 1%) would be winners, and thus, the probability of winning is 0.01 or 1%. A successful bet of $0.50 pays out $25 to the player.
Assume that for a given time period, 100,000 bets like the one described above were placed. How much money should the Maryland Lottery Agency expect to earn on an average from 100,000 bets?
Question 3 options:
$25,000
$50,000
$100,000
-$25,000
Question 4 (1 point)
A $20 bill, two $10 bills, three $5 bills, and four $1 bills are placed in a bag. If a bill is chosen at random, what is the expected value for the amount chosen? If you managed this game, how much would you charge people to play? Explain your reasoning.
Question 4 options:
You would want to charge people at least $20 to play.
You would want to charge people more than $5.90 to play.
You would want to charge people less than $5.90 to play.
You would want to charge people exactly $5.90 to play.
Question 5 (1 point)
As posted on the Maryland Lottery's website for its Pick 3 game, the chance of winning with a Front Pair bet is 0.01. A Front Pair bet is successful if the front pair of numbers you select match the Pick 3 number's first 2 digits. For example, a bet of 12__ would be a winner if the Pick 3 number is 120, 121, 122, and so on of any 3 digit number beginning with 12. In other words, 10 of the 1,000 possible Pick 3 numbers (or 1%) would be winners, and thus, the probability of winning is 0.01 or 1%. A successful bet of $0.50 pays out $25 to the player.
On average, how much does the Maryland lottery make on one bet?
Question 5 options:
-$0.25
$0.25
-$0.50
$0.50
Answers
GPT-4o mini
Let's go through each question one by one and calculate the expectations and other relevant details.
### Question 1
**Expected payout for a Front Pair bet:**
- Probability of winning (P(win)) = 0.01
- Payout for a successful bet = $25
- Cost to place a bet = $0.50
The expected payout (E) can be calculated as follows:
\[ E = (P(win) \times Payout) + (P(lose) \times Loss) \]
Where:
- P(lose) = 1 - P(win) = 0.99
- Loss = -$0.50 (the cost of the bet)
Plugging in the values:
\[ E = (0.01 \times 25) + (0.99 \times -0.50) \]
\[ E = 0.25 + (-0.495) \]
\[ E = 0.25 - 0.495 = -0.245 \]
So, rounding this gives:
**Expected payout ≈ -$0.25**
**Answer: -$0.25**
---
### Question 2
**Expected payoff for the card game:**
- Cost to play = $2
- Payout for winning (picking a spade) = $9
- Probability of winning (picking a spade) = 13 spades / 52 total cards = 1/4
- Probability of losing = 1 - 1/4 = 3/4
The expected payoff (E) is calculated as follows:
\[ E = (P(win) \times Payout) + (P(lose) \times Loss) \]
Where:
- Loss = -$2 (the cost of playing).
Plugging in the values:
\[ E = (0.25 \times (9 - 2)) + (0.75 \times (-2)) \]
\[ E = (0.25 \times 7) + (0.75 \times -2) \]
\[ E = 1.75 - 1.5 = 0.25 \]
**Expected payoff ≈ $0.25**
**Answer: You would win $0.25 on this game.**
---
### Question 3
**Expected earnings for the Maryland Lottery from 100,000 bets:**
- Probability of winning = 0.01
- Payout for winning = $25
- Cost of each bet = $0.50
Total amount wagered on 100,000 bets = 100,000 * $0.50 = $50,000.
Now calculate how many bets will win:
- Expected number of winning bets = 100,000 * 0.01 = 1,000
- Total payout to winners = 1,000 * $25 = $25,000
Now, to find the amount the Lottery makes:
\[ \text{Amount earned} = \text{Total wagered} - \text{Total payout} \]
\[ \text{Amount earned} = 50,000 - 25,000 = 25,000 \]
**Answer: $25,000**
---
### Question 4
**Expected value for the amount chosen:**
- Amount of bills:
- 1 x $20 = $20
- 2 x $10 = $20
- 3 x $5 = $15
- 4 x $1 = $4
- Total value of bills = $20 + $20 + $15 + $4 = $59
- Total number of bills = 1 + 2 + 3 + 4 = 10
Expected value (EV):
\[ EV = \frac{59}{10} = 5.9 \]
To determine how much to charge:
- You should charge just above $5.90 because if you charge less than that, you will start losing money.
**Answer: You would want to charge people more than $5.90 to play.**
---
### Question 5
**Expected earnings for the Maryland Lottery on one bet:**
As calculated in Question 1, the expected payout for a player is approximately -$0.25.
Hence, the lottery’s expected profit (how much they make on average from one bet):
**Answer: $0.25**
---
Summary of answers:
1. -$0.25
2. You would win $0.25 on this game.
3. $25,000
4. You would want to charge people more than $5.90 to play.
5. $0.25
### Question 1
**Expected payout for a Front Pair bet:**
- Probability of winning (P(win)) = 0.01
- Payout for a successful bet = $25
- Cost to place a bet = $0.50
The expected payout (E) can be calculated as follows:
\[ E = (P(win) \times Payout) + (P(lose) \times Loss) \]
Where:
- P(lose) = 1 - P(win) = 0.99
- Loss = -$0.50 (the cost of the bet)
Plugging in the values:
\[ E = (0.01 \times 25) + (0.99 \times -0.50) \]
\[ E = 0.25 + (-0.495) \]
\[ E = 0.25 - 0.495 = -0.245 \]
So, rounding this gives:
**Expected payout ≈ -$0.25**
**Answer: -$0.25**
---
### Question 2
**Expected payoff for the card game:**
- Cost to play = $2
- Payout for winning (picking a spade) = $9
- Probability of winning (picking a spade) = 13 spades / 52 total cards = 1/4
- Probability of losing = 1 - 1/4 = 3/4
The expected payoff (E) is calculated as follows:
\[ E = (P(win) \times Payout) + (P(lose) \times Loss) \]
Where:
- Loss = -$2 (the cost of playing).
Plugging in the values:
\[ E = (0.25 \times (9 - 2)) + (0.75 \times (-2)) \]
\[ E = (0.25 \times 7) + (0.75 \times -2) \]
\[ E = 1.75 - 1.5 = 0.25 \]
**Expected payoff ≈ $0.25**
**Answer: You would win $0.25 on this game.**
---
### Question 3
**Expected earnings for the Maryland Lottery from 100,000 bets:**
- Probability of winning = 0.01
- Payout for winning = $25
- Cost of each bet = $0.50
Total amount wagered on 100,000 bets = 100,000 * $0.50 = $50,000.
Now calculate how many bets will win:
- Expected number of winning bets = 100,000 * 0.01 = 1,000
- Total payout to winners = 1,000 * $25 = $25,000
Now, to find the amount the Lottery makes:
\[ \text{Amount earned} = \text{Total wagered} - \text{Total payout} \]
\[ \text{Amount earned} = 50,000 - 25,000 = 25,000 \]
**Answer: $25,000**
---
### Question 4
**Expected value for the amount chosen:**
- Amount of bills:
- 1 x $20 = $20
- 2 x $10 = $20
- 3 x $5 = $15
- 4 x $1 = $4
- Total value of bills = $20 + $20 + $15 + $4 = $59
- Total number of bills = 1 + 2 + 3 + 4 = 10
Expected value (EV):
\[ EV = \frac{59}{10} = 5.9 \]
To determine how much to charge:
- You should charge just above $5.90 because if you charge less than that, you will start losing money.
**Answer: You would want to charge people more than $5.90 to play.**
---
### Question 5
**Expected earnings for the Maryland Lottery on one bet:**
As calculated in Question 1, the expected payout for a player is approximately -$0.25.
Hence, the lottery’s expected profit (how much they make on average from one bet):
**Answer: $0.25**
---
Summary of answers:
1. -$0.25
2. You would win $0.25 on this game.
3. $25,000
4. You would want to charge people more than $5.90 to play.
5. $0.25