At a raffle, 1000 tickets are sold at $5 each. There are 20 prizes of $25, 5 prizes of $100, and 1 grand prize of $2000. Suppose you buy one ticket.

1. Use the table below to help you construct a probability distribution for all of the possible values of X and their probabilities

2. Find the expected value of X, and interpret it in the context of the game.
3. If you play in such a raffle 100 times, what is the expected net gain?
4. What ticket price (rounded to two decimal places) would make it a fair game?
5. Would you choose to play the game? In complete sentences, explain why or why not.
6. If you were organizing a raffle like this, how might you adjust the ticket prices and/or prize amounts in order to make the raffle more tempting while still raising at least $2000 for your organization?

To determine the probability of winning a specific prize, we need to know the total number of tickets sold and the number of tickets eligible to win that particular prize.

1. Total Number of Tickets Sold: The problem states that 1000 tickets were sold.

2. Number of Tickets Eligible for the Grand Prize ($2000): There is only 1 grand prize, so only 1 ticket is eligible to win it.

3. Number of Tickets Eligible for the $100 Prizes: There are 5 $100 prizes available, so 5 tickets are eligible to win them.

4. Number of Tickets Eligible for the $25 Prizes: There are 20 $25 prizes available, so 20 tickets are eligible to win them.

Now, let's calculate the probabilities step-by-step:

Step 1: Calculate the probability of winning the grand prize ($2000):
Probability of winning the grand prize = Number of tickets eligible for the grand prize / Total number of tickets sold
Probability of winning the grand prize = 1 / 1000
Probability of winning the grand prize = 1/1000 = 0.001 or 0.1%

Step 2: Calculate the probability of winning a $100 prize:
Probability of winning a $100 prize = Number of tickets eligible for a $100 prize / Total number of tickets sold
Probability of winning a $100 prize = 5 / 1000
Probability of winning a $100 prize = 1/200 = 0.005 or 0.5%

Step 3: Calculate the probability of winning a $25 prize:
Probability of winning a $25 prize = Number of tickets eligible for a $25 prize / Total number of tickets sold
Probability of winning a $25 prize = 20 / 1000
Probability of winning a $25 prize = 1/50 = 0.02 or 2%

So, the probability of winning the grand prize is 0.1%, the probability of winning a $100 prize is 0.5%, and the probability of winning a $25 prize is 2%.

If you buy one ticket, let's calculate your chances of winning a prize and the expected value of your ticket.

First, let's determine the probability of winning each available prize:

- There are 20 prizes of $25 each. The probability of winning one of these prizes is: 20/1000 = 1/50.
- There are 5 prizes of $100 each. The probability of winning one of these prizes is: 5/1000 = 1/200.
- Finally, there's 1 grand prize of $2000. The probability of winning this prize is: 1/1000.

Now, let's calculate the expected value of your ticket. The expected value is the weighted average of the possible prizes, weighted by their probabilities of winning.

Expected value = (Prize 1 * Probability of winning Prize 1) + (Prize 2 * Probability of winning Prize 2) + (Prize 3 * Probability of winning Prize 3) + ...

Expected value = ($25 * 1/50) + ($100 * 1/200) + ($2000 * 1/1000)

Expected value = ($0.5) + ($0.5) + ($2)

Expected value = $3

Therefore, the expected value of your ticket is $3. This means that in the long run, if you were to buy many tickets and enter this raffle multiple times, on average, you would expect to win $3 per ticket. However, it's important to note that this is just an average and does not guarantee you will win that amount exactly.

To summarize, if you buy one ticket, your probability of winning a prize is relatively low. The expected value of your ticket is $3.

OK. But what is your question?