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Chapter 5: Problem 8

For a fundraiser, 1000 raffle tickets are sold and the winner is chosen at random. There is only one prize, 500 dollar in cash. You buy one ticket. (a) What is the probability you will win the prize of 500 dollar? (b) Your expected earnings can be found by multiplying the value of the prize by the probability you will win the prize. What are your expected earnings? (c) Interpretation If a ticket costs 2 dollar, what is the difference between your "costs" and "expected earnings"? How much are you effectively contributing to the fundraiser?

Short Answer

Expert verified
(a) \( \frac{1}{1000} \); (b) 0.5 dollars; (c) You contribute 1.5 dollars to the fundraiser.

Step by step solution

01

Calculate the probability of winning

The probability of winning the raffle is calculated by dividing the number of winning tickets by the total number of tickets. Since there is only one prize, the only winning ticket is the one you bought. Hence, the probability of winning is \( \frac{1}{1000} \).
02

Calculate expected earnings

The expected earnings are calculated by multiplying the probability of winning by the value of the prize. This is calculated as follows: \( \frac{1}{1000} \times 500 = 0.5 \). Therefore, the expected earnings are 0.5 dollars.
03

Calculate difference between cost and expected earnings

The cost of purchasing one raffle ticket is 2 dollars. The difference between this cost and your expected earnings (0.5 dollars) is calculated by subtracting the expected earnings from the cost: \( 2 - 0.5 = 1.5 \).
04

Determine contribution to the fundraiser

The difference calculated in Step 3 represents your contribution to the fundraiser. In this case, you contribute 1.5 dollars to the fundraiser.

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Key Concepts

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

Understanding Raffle Tickets
Raffle tickets are a simple form of lottery, where participants purchase a chance to win a prize. Each ticket has an equal probability of winning, meaning that if there are 1,000 tickets sold, and only one prize, the chance of any single ticket winning is 1 in 1,000. This probability is expressed mathematically as \( \frac{1}{1000} \).

For participants, the appeal of raffle tickets lies in the small cost for a potentially large reward. In our example, the cost for a ticket is just 2 dollars, while the prize is 500 dollars. This setup creates both excitement and a worthwhile cause, as funds raised from selling the tickets generally support local communities or charities. Participants thus not only purchase a chance to win but also often contribute to a noble cause.
Calculating Expected Value
The expected value is a concept in probability that offers a measure of the center of a random variable's distribution. When applying this to raffle tickets, the expected value represents what each participant can "expect" to win on average if the game were to be repeated many times.

To calculate expected value, you multiply the probability of winning by the prize amount. In this instance, the expected winnings equal \( \frac{1}{1000} \times 500 \), which results in 0.5 dollars for each ticket. This means that while you pay 2 dollars per ticket, mathematically, you can expect to "win" back 0.5 dollars. This estimation doesn't change the outcome of the actual raffle but helps participants understand the average value of their ticket in the long term.
Analyzing Fundraiser Contribution
When you buy a raffle ticket, your payment typically serves two purposes: a chance at winning a prize and supporting a fundraising cause. In our exercise, by purchasing a ticket for 2 dollars, you directly affect both these avenues.

Given the expected earnings are 0.5 dollars per ticket, the remaining 1.5 dollars paid for the ticket contributes directly to the fundraiser's goal. This difference highlights the dual role that raffle tickets play鈥攏ot only as a game of chance but also as a tool for charitable giving. Therefore, even if not winning the prize, participants contribute to a larger cause, which is often the true purpose behind raffles.

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Most popular questions from this chapter

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