Let's calculate the expected value of playing this game step by step.

Step 1: Understand the outcomes

The die has six faces, numbered from 1 to 6. The outcomes can be divided into:

• Even numbers: 2, 4, 6
• Odd numbers: 1, 3, 5

Step 2: Define the possible outcomes

• If the die lands on an even number, you win 7 times the rolled amount.
• If the die lands on an odd number, you lose 10 times the rolled amount.

Step 3: Calculate the winnings/losses

We now calculate the amount you win or lose for each roll:

• If you roll a 2 (even): win $7 \times 2 = 14$.
• If you roll a 4 (even): win $7 \times 4 = 28$.
• If you roll a 6 (even): win $7 \times 6 = 42$.
• If you roll a 1 (odd): lose $10 \times 1 = 10$.
• If you roll a 3 (odd): lose $10 \times 3 = 30$.
• If you roll a 5 (odd): lose $10 \times 5 = 50$.

Step 4: Find the probabilities

Each face of the die has an equal probability of landing, which is $\frac{1}{6}$.

Step 5: Compute the expected value

The expected value $E$ is calculated by summing the products of each outcome and its respective probability:

$E = \left( \frac{1}{6} \times 14 \right) + \left( \frac{1}{6} \times 28 \right) + \left( \frac{1}{6} \times 42 \right) + \left( \frac{1}{6} \times (-10) \right) + \left( \frac{1}{6} \times (-30) \right) + \left( \frac{1}{6} \times (-50) \right)$

Now, simplify and calculate:

$E = \frac{1}{6} \left( 14 + 28 + 42 - 10 - 30 - 50 \right)$ $E = \frac{1}{6} \times (-6)$ $E = -1$

Step 6: Conclusion

The expected value of playing this game is -1. This means that, on average, you lose 1 unit each time you play the game.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you calculate expected value for games involving both wins and losses?
2. How would the expected value change if the winnings or losses were different?
3. What does the expected value tell you about a game over a large number of plays?
4. How can you use expected value to decide whether a game is fair?
5. How does the probability of an outcome affect the expected value?

Tip: When calculating expected value, always break down the possible outcomes, and multiply each by its probability to find the overall expected result.