In a certain state's lottery, 45 balls numbered 1 through 45 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order in which the numbers are drawn does not matter.

Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket. Write your answer as a reduced fraction.

P𝑃(win) = 18145060​Correct  

A single lottery ticket costs $2. Compute the Expected Value, to the state, if 10,000 lottery tickets are sold. Round your answer to the nearest dollar.

Answer: $Incorrect

A single lottery ticket costs $2. Compute the Expected Value, to you, if you purchase 10,000 lottery tickets. Round your answer to the nearest dollar.

Answer: $

Let's go through the steps to solve these problems related to the lottery scenario.

### 1. Compute the Probability of Winning

To win the lottery, the player must correctly choose all 6 numbers out of 45. The total number of ways to choose 6 numbers out of 45 is given by the combination formula $\binom{45}{6}$.

\[ \binom{45}{6} = \frac{45!}{6!(45-6)!} = \frac{45!}{6! \times 39!} \]

First, let's compute $\binom{45}{6}$:

\[ \binom{45}{6} = \frac{45 \times 44 \times 43 \times 42 \times 41 \times 40}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

Calculating the numerator:

\[ 45 \times 44 \times 43 \times 42 \times 41 \times 40 = 58623876440 \]

And the denominator:

\[ 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

So,

\[ \binom{45}{6} = \frac{58623876440}{720} = 8132160 \]

Thus, the probability of winning the lottery with a single ticket is:

\[ P(\text{win}) = \frac{1}{8132160} \]

### 2. Compute the Expected Value to the State if 10,000 Tickets are Sold

Each ticket costs $2, so the total revenue from selling 10,000 tickets is:

\[ \text{Total revenue} = 10,000 \times 2 = 20,000 \text{ dollars} \]

Since the probability of winning with a single ticket is $\frac{1}{8132160}$, the expected number of winners if 10,000 tickets are sold is:

\[ \text{Expected number of winners} = 10,000 \times \frac{1}{8132160} \approx 0.00123 \]

The prize for winning is $1,000,000, so the expected payout by the state is:

\[ \text{Expected payout} = 0.00123 \times 1,000,000 \approx 1230 \text{ dollars} \]

Therefore, the expected profit for the state is:

\[ \text{Expected profit} = \text{Total revenue} - \text{Expected payout} = 20,000 - 1230 = 18,770 \text{ dollars} \]

Rounded to the nearest dollar:

\[ \text{Expected profit for the state} = 18,770 \text{ dollars} \]

### 3. Compute the Expected Value to You if You Purchase 10,000 Tickets

If you purchase 10,000 tickets, you spend:

\[ \text{Total cost} = 10,000 \times 2 = 20,000 \text{ dollars} \]

The expected number of wins is still:

\[ 10,000 \times \frac{1}{8132160} \approx 0.00123 \]

The expected winnings are:

\[ \text{Expected winnings} = 0.00123 \times 1,000,000 \approx 1230 \text{ dollars} \]

Therefore, the expected net value to you is:

\[ \text{Expected net value} = \text{Expected winnings} - \text{Total cost} = 1230 - 20,000 = -18,770 \text{ dollars} \]

Rounded to the nearest dollar:

\[ \text{Expected net value to you} = -18,770 \text{ dollars} \]

### Summary of Results

1. **Probability of winning the lottery with a single ticket:**
   \[ \frac{1}{8132160} \]

2. **Expected profit to the state if 10,000 tickets are sold:**
   \[ 18,770 \text{ dollars} \]

3. **Expected net value to you if you purchase 10,000 tickets:**
   \[ -18,770 \text{ dollars} \]

Would you like further details or have any questions?

Here are 5 questions for you to explore:

1. How do you compute combinations for a given set of elements?
2. What is the difference between expected value and actual value?
3. How can understanding probability improve decision-making in games of chance?
4. What are the key factors that affect the expected value in lotteries?
5. How can you use probability to evaluate financial decisions?

**Tip:** When dealing with expected values, always consider both the potential gains and the costs involved to understand the net impact of your decisions.

Explore the probability of winning a lottery by correctly choosing 6 numbers out of 45. Understand the expected value for both the state and individual players when purchasing multiple tickets. Includes calculations and insights on financial outcomes.

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