Step 1: Calculate the probability of winning

The lottery involves picking 6 numbers between 0 and 9. Each number can independently take on any value from 0 to 9, which means there are $10^6$ possible combinations of numbers.

$\text{Total possible combinations} = 10^6 = 1,000,000$

The probability of picking the correct 6 numbers is therefore:

$P(\text{win}) = \frac{1}{1,000,000}$

Step 2: Expected value calculation

The expected value ($E$) of playing the lottery is calculated as:

$E = \text{Probability of winning} \times \text{Grand prize} - \text{Cost of ticket}$

Let the grand prize be denoted as $G$. The expected value becomes:

$E = \frac{1}{1,000,000} \cdot G - 1,000$

For the expected value to be positive, we set $E > 0$:

$\frac{1}{1,000,000} \cdot G - 1,000 > 0$

$\frac{1}{1,000,000} \cdot G > 1,000$

$G > 1,000 \times 1,000,000$

$G > 1,000,000,000$

Step 3: Answering the questions

1. How much would the grand prize need to be for a positive expected value? The grand prize would need to be greater than $1 billion for you to have a positive expected value.

2. Would you play this game if the grand prize was $10 billion? If the grand prize is $G = 10,000,000,000$, the expected value is:

   $E = \frac{1}{1,000,000} \cdot 10,000,000,000 - 1,000$

   $E = 10,000 - 1,000 = 9,000$

   Since the expected value is positive ($E = 9,000$), it would make sense to play this game if you can afford the $1,000 ticket cost and are willing to take the risk.

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Follow-up Questions

1. What happens to the expected value if the cost of the ticket increases?
2. How would adding more numbers to the lottery (e.g., picking 7 numbers) change the expected value?
3. What if the lottery allowed you to buy multiple tickets—how does that affect your chances and expected value?
4. How does the concept of risk aversion influence the decision to play this game?
5. Can taxes on lottery winnings alter whether this game is worth playing?

Tip: Always consider the risk versus reward ratio when making decisions involving high variability, especially with low probabilities of success.