A lottery costs 1 dollar to play. Bella must pick four digits in order from 0 to 9 and duplicates are allowed. If she wins, the prize is 5,000 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.

Step 1 ：The lottery costs 1 dollar to play. Bella must pick four digits in order from 0 to 9 and duplicates are allowed. If she wins, the prize is 5,000 dollars. We are asked to calculate the expected value of the lottery and determine whether the lottery is a fair game. If the game is not fair, we need to determine a price for playing the game that would make it fair.

Step 2 ：The expected value of a game is calculated by multiplying each possible outcome by the probability of that outcome, and then summing these values. In this case, the possible outcomes are winning $5,000 and losing $1.

Step 3 ：The probability of winning is the number of winning combinations divided by the total number of combinations. Since there are 10 digits and 4 are chosen with replacement, there are \(10^4\) total combinations. There is only 1 winning combination, so the probability of winning is \(\frac{1}{10^4}\). The probability of losing is 1 minus the probability of winning.

Step 4 ：A game is fair if the expected value is 0. If the game is not fair, a fair price can be calculated by setting the expected value to 0 and solving for the cost of the game.

Step 5 ：By calculating, we find that the probability of winning is 0.0001, the expected value of the game is -0.4999 dollars. This means the game is not fair. The player is expected to lose money in the long run.

Step 6 ：A fair price for the game would be 0.5 dollars. This is the price that would make the expected value of the game 0, meaning the player would break even in the long run.

Step 7 ：Final Answer: The expected value of the lottery is -0.4999 dollars. The lottery is not a fair game. A fair price for the lottery would be \(\boxed{0.5}\) dollars.