The question in the image states:

"A lottery costs 1 dollar to play. Bella must pick two digits in order from 0 to 9 and duplicates are allowed. If she wins, the prize is 80 dollars. First calculate the expected value of the lottery. Determine whether the lottery is a fair game. If the game is not fair, determine a price for playing the game that would make it fair."

Let's break this down step by step:

Step 1: Calculate the total possible outcomes

Bella picks two digits in order, and duplicates are allowed.

• For each digit, there are 10 options (0 through 9).
• The total number of outcomes is: $10 \times 10 = 100$

Step 2: Probability of winning

The probability of winning is 1 favorable outcome out of 100 total outcomes: $P(\text{win}) = \frac{1}{100}$

Step 3: Expected value of the lottery

The expected value (EV) is calculated as: $\text{EV} = (\text{probability of win}) \times (\text{winning amount}) + (\text{probability of loss}) \times (\text{losing amount})$ Substitute the values:

• Winning amount = $80
• Losing amount = $-1$ (the cost of playing the game)
• Probability of losing = $1 - \frac{1}{100} = \frac{99}{100}$

$\text{EV} = \left(\frac{1}{100} \times 80\right) + \left(\frac{99}{100} \times (-1)\right)$ $\text{EV} = 0.8 - 0.99 = -0.19$

So, the expected value is $-0.19$ dollars, meaning Bella loses 19 cents on average for every game played.

Step 4: Determine if the lottery is fair

A lottery is considered fair if the expected value is 0, meaning there is no financial gain or loss for the player on average.
Since the expected value is negative ($-0.19$), the lottery is not fair.

Step 5: Fair price for the lottery

To make the game fair, adjust the cost of playing such that the expected value equals 0. Let the fair price of playing be $x$.

$\text{EV} = \left(\frac{1}{100} \times 80\right) + \left(\frac{99}{100} \times (-x)\right) = 0$ $0.8 - 0.99x = 0$ $0.99x = 0.8$ $x = \frac{0.8}{0.99} \approx 0.81$

The fair price for playing the game is approximately 81 cents.

Would you like a detailed explanation of any step? Here are some related questions:

1. What is the significance of the expected value in gambling?
2. How do we calculate probabilities for different types of lotteries?
3. Can you explain how duplicates in digit selection affect probabilities?
4. What if the prize amount were different—how would that affect fairness?
5. How could a business use expected value to design fairer lotteries?

Tip: When solving problems involving probabilities and fairness, always verify your assumptions about total outcomes and payouts carefully.