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Chapter 4: Problem 9

You purchase a raffle ticket to help out a charity. The raffle ticket costs \$5. The charity is selling 2000 tickets. One of them will be drawn and the person holding the ticket will be given a prize worth \(\$ 4000\). Compute the expected value for this raffle.

Short Answer

Expert verified
The expected value of the raffle ticket is \\$-3\\$.

Step by step solution

01

Define Probability and Outcomes

First, we need to define the probability of winning and losing, as well as the outcomes for each. The probability of winning the raffle is \( \frac{1}{2000} \), since only one ticket wins out of 2000. The probability of losing is \( \frac{1999}{2000} \) since there are 1999 tickets that do not win.
02

Calculate Expected Value for Winning

The outcome if you win is a prize of \(4000 minus the \)5 cost of the ticket, resulting in a net gain of \(4000 - 5 = 3995\) dollars. The expected value of winning is therefore given by multiplying the net gain by the probability of winning: \( \frac{1}{2000} \times 3995 = 1.9975 \) dollars.
03

Calculate Expected Value for Losing

The outcome if you lose is the complete loss of the $5 cost of the ticket. Therefore, the expected value of losing is the probability of losing multiplied by this loss: \( \frac{1999}{2000} \times (-5) = -4.9975 \) dollars.
04

Calculate Total Expected Value

The expected value of purchasing the ticket is the sum of the expected value of winning and the expected value of losing: \( 1.9975 + (-4.9975) = -3 \).
05

Conclusion

The expected value of the raffle ticket is negative, meaning on average, you would lose $3 in this raffle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is a mathematical framework used to quantify uncertainty. In simple terms, it allows us to predict how likely an event is to occur. In the context of the raffle exercise, we apply probability to estimate the chances of winning or losing when buying a raffle ticket.

Here are the basic concepts of probability:
  • **Probability of an Event**: This refers to how likely the event is to happen. It's expressed as a ratio of favorable outcomes to the total number of possible outcomes. For instance, the probability of winning a raffle where one ticket wins out of 2000 is \( \frac{1}{2000} \).
  • **Complementary Probability**: This is the probability of the event not happening. For example, if the probability of winning is \( \frac{1}{2000} \), the complementary probability (losing the raffle) is \( \frac{1999}{2000} \).
By understanding these concepts, you can better gauge the risks and potential rewards involved in probabilistic situations like raffles.
Expected Value
The expected value is a core concept in probability theory used to determine the mean outcome of a probabilistic event over time. It's particularly useful when evaluating risks in lotteries, investments, and games of chance.

Here's how expected value works:
  • **Net Outcome Calculation**: For each potential outcome, you calculate the net gain or loss. For example, in the raffle, if you win, your net gain is \( 4000 - 5 = 3995 \) dollars, whereas losing results in a loss of \( 5 \) dollars.
  • **Probability Weighted Sum**: Multiply each net outcome by its probability. This gives the contribution of each outcome to the overall expected value. For our raffle, the expected value of winning is \( \frac{1}{2000} \times 3995 = 1.9975 \) dollars, and for losing, it's \( \frac{1999}{2000} \times (-5) = -4.9975 \) dollars.
  • **Summation for Total Expected Value**: Add the values from all possible outcomes. For this raffle, it means summing \( 1.9975 \) and \( -4.9975 \) to find the total expected value, which results in \( -3 \) dollars.
Understanding expected value helps to recognize that even though the raffle offers a big prize, the average outcome over time is financially unwise.
Raffle Mathematics
Raffle mathematics involves understanding the dynamics of buying lottery or raffle tickets and calculating outcomes. Given that raffles are games of chance, they provide a practical application of probability and expected value concepts.

To successfully navigate raffle mathematics, consider the following points:
  • **Cost vs. Prize Potential**: Always evaluate the cost of participation against the potential prize. In our example, the ticket cost is \( 5 \), while the prize is \( 4000 \).
  • **Probability of Winning**: With 2000 tickets, the chances of winning are very slim, \( \frac{1}{2000} \).
  • **Expected Value Interpretation**: Negative expected value, such as \( -3 \) dollars in our example, implies a loss over time. As such, regular purchases might lead to a cumulative financial loss.
Even if winning can be attractive, understanding raffle mathematics means accepting that statistically, buying a ticket is typically not a sound financial decision.

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You purchase a raffle ticket to help out a charity. The raffle ticket costs \(\$ 10 .\) The charity is selling 2000 tickets. One of them will be drawn and the person holding the ticket will be given a prize worth \(\$ 8000\). Compute the expected value for this raffle.

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