Five thousand tickets are sold at​ $1 each for a charity raffle. Tickets are to be drawn at random and monetary prizes awarded as​ follows: 1 prize of

​$700700​,

3 prizes of

​$200200​,

5 prizes of

​$1010​,

and 20 prizes of​ $5. What is the expected value of this raffle if you buy 1​ ticket?

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Part 1

Let X be the random variable for the amount won on a single raffle ticket.

​E(X)equals=enter your response here

dollars ​(Round to the nearest cent as​ needed.)

To find the expected value of the raffle, we need to calculate the expected value $ E(X) $ of the random variable $ X $, which represents the amount won on a single raffle ticket. 

The expected value is calculated as follows:

\[ E(X) = \sum (x_i \cdot p_i) \]

where $ x_i $ is the value of each prize, and $ p_i $ is the probability of winning that prize.

Here are the given prizes and their probabilities:

- 1 prize of $700: $\frac{1}{5000}$
- 3 prizes of $200: $\frac{3}{5000}$
- 5 prizes of $10: $\frac{5}{5000}$
- 20 prizes of $5: $\frac{20}{5000}$
- The rest of the tickets (4971) win $0: $\frac{4971}{5000}$

Let's calculate each part:

1. For the $700 prize:
\[ 700 \times \frac{1}{5000} = 0.14 \]

2. For the three $200 prizes:
\[ 200 \times \frac{3}{5000} = 0.12 \]

3. For the five $10 prizes:
\[ 10 \times \frac{5}{5000} = 0.01 \]

4. For the twenty $5 prizes:
\[ 5 \times \frac{20}{5000} = 0.02 \]

5. For the $0 prize:
\[ 0 \times \frac{4971}{5000} = 0 \]

Adding these together gives the expected value:

\[ E(X) = 0.14 + 0.12 + 0.01 + 0.02 + 0 = 0.29 \]

Therefore, the expected value of the raffle if you buy 1 ticket is $ \boxed{0.29} $ dollars.

Do you want details or have any questions?

Here are 5 more questions you might ask:
1. How do I calculate the variance of the raffle ticket winnings?
2. What is the probability of not winning any prize in this raffle?
3. How would the expected value change if the number of tickets sold was different?
4. How can I use expected value in other types of lotteries or games of chance?
5. What does the expected value tell me about the fairness of the raffle?

Tip: When calculating expected values, ensure all possible outcomes and their probabilities are accounted for to achieve accurate results.

Learn how to calculate the expected value of a charity raffle ticket where prizes include $700, $200, $10, and $5. Understand the probability distribution and expected winnings per ticket. Suitable for advanced high school students.

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