To determine the expected value for the carnival game, we need to take into account the charge for playing the game, the payout for winning, and the probability of winning. Let's break this down step by step:
### Step 1: Define the Variables
1. Charge per game (Revenue): \[tex]$2
2. Payout for a win: \$[/tex]5
3. Probability of winning: 0.1 (or 10%)
### Step 2: Calculate the Expected Payout Per Game
The expected payout is calculated by multiplying the probability of winning by the payout for a win:
[tex]\[ \text{Expected Payout} = \text{Probability of Win} \times \text{Payout for Win} \][/tex]
[tex]\[ \text{Expected Payout} = 0.1 \times 5 = 0.5 \][/tex]
### Step 3: Calculate the Expected Revenue Per Game
The expected revenue is simply the charge per game:
[tex]\[ \text{Expected Revenue} = \$2 \][/tex]
### Step 4: Calculate the Expected Net Gain Per Game
The expected net gain (expected value) for the game is the difference between the expected revenue and the expected payout:
[tex]\[ \text{Expected Value} = \text{Expected Revenue} - \text{Expected Payout} \][/tex]
[tex]\[ \text{Expected Value} = 2 - 0.5 = 1.5 \][/tex]
### Summary of Results:
- Expected Payout: \[tex]$0.5
- Expected Revenue: \$[/tex]2
- Expected Value: \[tex]$1.5
Therefore, the expected value for the game is \$[/tex]1.5. This means that on average you can expect to gain \[tex]$1.5 each time someone plays your game, not lose \$[/tex]0.15 as stated in the instructions.
### Fairness of the Game
Mathematically, a game is considered "fair" if the expected value is zero, meaning players and the game organizer (you) neither win nor lose money on average over time:
[tex]\[ \text{Expected Value} = \$0 \][/tex]
In this case, the expected value is \[tex]$1.5, which indicates that, on average, you will be making a profit of \$[/tex]1.5 for each game played.
Thus, the game is not fair for the players as they have a negative expected value, whereas it is highly favorable for you as the game organizer.
### Conclusion
- The expected value for the game is \[tex]$1.5.
- On average, you can expect to gain \$[/tex]1.5 each time someone plays your game.
- Mathematically, the game is not considered fair for the players.
