To determine if you can expect to break even in the long run and whether the game is fair, we need to analyze the expected value of the game.

The fairness of a game is determined by comparing the cost to play with the expected value (the average outcome if the game is played many times).

Here's how we typically approach this:

1. Find the expected value (EV) of the game. This is the sum of all possible outcomes, each multiplied by its probability.
2. Compare the expected value with the cost of playing the game.
   • If the expected value is equal to the cost, the game is fair.
   • If the expected value is greater than the cost, the player can expect to profit.
   • If the expected value is less than the cost, the player is expected to lose in the long run.

Steps to complete the analysis:

• Do you have the details of the game, such as the possible prizes or outcomes and their probabilities?

Once I have that information, I can calculate the expected value and answer both questions clearly.

Follow-up questions:

1. What are the possible prizes or outcomes of the game and their probabilities?
2. Have you played the game multiple times, and what were the results?
3. What kind of game are you analyzing (e.g., lottery, dice, etc.)?
4. Are there any external factors or additional costs to playing the game (besides the $3.00)?
5. How does this game compare to other games in terms of fairness or house advantage?

Tip: Understanding expected value is crucial in determining the fairness of any probabilistic game. The higher the expected value compared to the cost, the better the game is for the player.