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When participating in a lottery, understanding the probability of winning is crucial. This probability tells us how likely we are to win a specific prize. In a lottery setup like the one we're analyzing, the probability is determined by the number of tickets compared to the number of prizes available. For example, if there is only one prize and 100 tickets, the chance to win that prize is \( \frac{1}{100} \) because there's only one winning ticket out of 100.

• For the \(500 prize, the chance of winning is \( \frac{1}{100} \).
• For one of the \)100 prizes, since there are two available, the chance becomes \( \frac{2}{100} \).
• For one of the $25 prizes, with four available, the probability is \( \frac{4}{100} \).

So, each ticket you buy gives you these odds for each respective prize. Understanding these probabilities helps manage expectations and informs you about your potential chances before entering the lottery.

A lottery setup outlines all the details about how the lottery is organized. This includes the total number of tickets available, the cost per ticket, and the prizes that can be won. All of these elements work together to shape the entire lottery experience.
In our situation:

• Total tickets: 100
• Cost per ticket: $10
• Prizes: one $500 prize, two $100 prizes, and four $25 prizes

Such clear details are essential as they directly affect the expected results of participating. The essence of understanding the setup means knowing not just what you might win, but how likely you are to win and how much you need to spend for a chance. This setup will dictate how the probability calculations, and ultimately the expected value, are formulated.

Expected gain or loss is a concept used to predict the average financial outcome from participating in an event like a lottery, over the long term. This is determined by calculating the expected value (EV) and then considering the initial cost of participation.
To find the expected value for this lottery, multiply each prize by its probability of occurrence:

• EV for \(500 prize: \( 500 \times \frac{1}{100} = 5 \)
• EV for \)100 prizes: \( 100 \times \frac{2}{100} = 2 \)
• EV for \(25 prizes: \( 25 \times \frac{4}{100} = 1 \)

Adding these gives a total expected value of \( 5 + 2 + 1 = 8 \).
Since each ticket costs \)10, the expected gain or loss is calculated by subtracting the ticket cost from the expected value: \[ EV - \text{Cost} = 8 - 10 = -2 \]This negative result means that, on average, you would lose $2 per ticket purchased. This insight helps individuals make informed decisions about whether or not it is worth participating in the lottery from a financial perspective.