Lucy is playing a game of chance in which she rolls a number cube with sides numbered from 1 to 6. The number cube is fair, so a side is rolled at random.

This game is this: Lucy rolls the number cube once. She wins $1 if a 1 is rolled, $2 if a 2 is rolled, $3 if a 3 is rolled, and $4 if a 4 is rolled. She loses $4.25 if a 5 or 6 is rolled. (a)Find the expected value of playing the game. Dollars.

To find the expected value of Lucy's game, we need to calculate the average outcome she can expect per roll of the number cube, considering all possible results and their probabilities.

Given the probability distribution of the game:
- Lucy wins $ \$1 $ if a 1 is rolled (probability $ \frac{1}{6} $)
- Lucy wins $ \$2 $ if a 2 is rolled (probability $ \frac{1}{6} $)
- Lucy wins $ \$3 $ if a 3 is rolled (probability $ \frac{1}{6} $)
- Lucy wins $ \$4 $ if a 4 is rolled (probability $ \frac{1}{6} $)
- Lucy loses $ \$4.25 $ if a 5 or 6 is rolled (combined probability $ \frac{2}{6} = \frac{1}{3} $)

### Calculating the Expected Value (E)
The expected value $ E $ can be calculated using the formula:
\[
E = \sum [x_i \times P(x_i)]
\]
Where $ x_i $ is the value of each outcome and $ P(x_i) $ is the probability of each outcome.

For Lucy's game, the expected value is:
\[
E = (1 \times \frac{1}{6}) + (2 \times \frac{1}{6}) + (3 \times \frac{1}{6}) + (4 \times \frac{1}{6}) + (-4.25 \times \frac{1}{3})
\]

Let's compute this value.The expected value of playing the game, given Lucy's conditions with the number cube, is $ \$0.25 $. This means that on average, Lucy can expect to gain 25 cents per roll of the cube over the long run.

Would you like more details or have any questions about this calculation?

Here are some related questions to consider:
1. How does the expected value change if the loss amount is increased or decreased?
2. What would happen to the expected value if another winning or losing number is added?
3. How many games would Lucy need to play to reach a certain profit level based on this expected value?
4. Can you explain how probability affects decision-making in games of chance like this one?
5. What is the probability that Lucy wins money in a single game?
6. How would the game's fairness change if the payout for winning numbers were adjusted?
7. What are the odds against Lucy for each roll of the dice?
8. Could you simulate this game using a computer program to see actual outcomes over many rolls?

**Tip**: When calculating expected values in games involving wins and losses, consider how the risk of losing might psychologically impact a player's decision to participate, despite the mathematical expectation.

Learn how to calculate the expected value of a game involving rolling a number cube. Understand the probabilities and outcomes associated with each roll, and find out the average earnings per roll. Suitable for high school students studying probability and game theory.

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