/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Basic Computation: Expected Valu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 6

Basic Computation: Expected Value For a fundraiser, 1000 raffle tickets are sold and the winner is chosen at random. There is only one prize, \(\$ 500\) in cash. You buy one ticket. (a) What is the probability you will win the prize of \(\$ 500\) ? (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 \(\$ 2\), what is the difference between your "costs" and "expected earnings"? How much are you effectively contributing to the fundraiser?

Short Answer

Expert verified
Probability of winning is \(\frac{1}{1000}\). Expected earnings are \$0.50. Contribution to the fundraiser is \$1.50.

Step by step solution

01

Calculate the Probability of Winning

The total number of tickets sold is 1000 and you have purchased 1 ticket. Therefore, the probability of winning the prize can be calculated using the formula \( \text{Probability of Winning} = \frac{\text{Number of Your Tickets}}{\text{Total Number of Tickets}} \). Thus, the probability of winning is \( \frac{1}{1000} \).
02

Determine the Expected Earnings

The expected earnings can be calculated by multiplying the amount of the prize by the probability of winning. The formula is \( \text{Expected Earnings} = \text{Prize Amount} \times \text{Probability of Winning} \). Substituting the values, we get \( \text{Expected Earnings} = 500 \times \frac{1}{1000} = 0.5 \). Therefore, your expected earnings are \$0.50.
03

Calculate the Difference between Costs and Expected Earnings

The cost of the ticket is \\(2. Your expected earnings are \\)0.50. Therefore, the difference between your costs and expected earnings can be calculated as \( \text{Difference} = \text{Cost} - \text{Expected Earnings} \). Thus, the difference is \( 2 - 0.5 = 1.5 \).
04

Interpretation of Contribution to Fundraiser

Since the difference between what you pay (\\(2.00) and your expected earnings (\\)0.50) is \\(1.50, this amount effectively represents your contribution to the fundraiser. You are contributing \\)1.50 to the fundraiser.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability
Probability is a measure of the likelihood that a certain event will occur. When participating in a raffle, the probability of winning is an essential factor to consider. You calculate it by taking the number of favorable outcomes and dividing it by the total number of possible outcomes.

In the context of a raffle with 1,000 tickets where each ticket is randomly drawn, if you buy one ticket, your chance of winning is calculated as:
  • Number of Your Tickets: 1
  • Total Number of Tickets: 1,000
  • Probability of Winning: \( \frac{1}{1000} \)
The probability, expressed as a fraction, gives you a clear view of how likely you are to win the prize.
Raffle
A raffle is a popular fundraising event where participants purchase tickets for the chance to win prizes. Each ticket has a unique number, giving the holder a potential chance at winning the announced rewards. Raffles are often used by schools, charities, and organizations as an effective way to collect funds.

The key components of a raffle include:
  • Tickets: Purchased by participants.
  • Prize: Rewards for the lucky winner.
  • Drawing: Random selection of a winning ticket.
Raffles create excitement and engage people in fundraising activities, making them a favored choice for raising money.
Fundraiser
A fundraiser is an event or campaign aimed at collecting money for a particular cause or project. Fundraisers can take various forms, such as auctions, bake sales, or raffles like the one mentioned in the exercise. The primary objective is to gather financial support from the community.

There are several purposes for organizing a fundraiser:
  • Supporting charitable causes.
  • Financing projects or operations.
  • Raising awareness about specific issues.
Each fundraising campaign often involves thoughtful planning and engagement to ensure it meets its financial goals effectively.
Contribution to Fundraiser
Your contribution to a fundraiser is the amount you effectively give after considering the cost of participation and any winnings. Using the raffle example, you contribute more than just the cost of a ticket if your expected earnings are less than what you paid.

To assess your contribution:
  • Cost of Ticket: \(2)
  • Expected Earnings: \)0.50
  • Net Contribution: \(2 - 0.50 = 1.50 \)
Understanding this difference helps participants realize how their payments aid the actual fundraising goal, adding clarity to how much is truly supporting the cause.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Binomial Probabilities: Coin Flip A fair quarter is flipped three times. For each of the following probabilities, use the formula for the binomial distribution and a calculator to compute the requested probability. Next, look up the probability in Table 3 of Appendix II and compare the table result with the computed result. (a) Find the probability of getting exactly three heads. (b) Find the probability of getting exactly two heads. (c) Find the probability of getting two or more heads. (d) Find the probability of getting exactly three tails.

Statistical Literacy What does the random variable for a binomial experiment of \(n\) trials measure?

Expected Value: Life Insurance Sara is a 60 -year-old Anglo female in reasonably good health. She wants to take out a \(\$ 50,000\) term (that is, straight death benefit) life insurance policy until she is \(65 .\) The policy will expire on her 65 th birthday. The probability of death in a given year is provided by the Vital Statistics Section of the Statistical Abstract of the United States (116th Edition). \begin{tabular}{l|ccccc} \hline\(x=\) age & 60 & 61 & 62 & 63 & 64 \\ \hline\(P\) (death at this age) & \(0.00756\) & \(0.00825\) & \(0.00896\) & \(0.00965\) & \(0.01035\) \\ \hline \end{tabular} Sara is applying to Big Rock Insurance Company for her term insurance policy. (a) What is the probability that Sara will die in her 60 th year? Using this probability and the \(\$ 50,000\) death benefit, what is the expected cost to Big Rock Insurance? (b) Repeat part (a) for years \(61,62,63\), and 64 . What would be the total expected cost to Big Rock Insurance over the years 60 through \(64 ?\) (c) Interpretation If Big Rock Insurance wants to make a profit of \(\$ 700\) above the expected total cost paid out for Sara's death, how much should it charge for the policy? (d) Interpretation If Big Rock Insurance Company charges \(\$ 5000\) for the policy, how much profit does the company expect to make?

Quota Problem: Archaeology An archaeological excavation at Burnt Mesa Pueblo showed that about \(10 \%\) of the flaked stone objects were finished arrow points (Source: Bandelier Archaeological Excavation Project: Summer 1990 Excavations at Burnt Mesa Pueblo, edited by Kohler, Washington State University). How many flaked stone objects need to be found to be \(90 \%\) sure that at least one is a finished arrow point? Hint: Use a calculator and note that \(P(r \geq 1) \geq 0.90\) is equivalent to \(1-P(0) \geq 0.90\), or \(P(0) \geq 0.10 .\)

Security: Burglar Alarms A large bank vault has several automatic burglar alarms. The probability is \(0.55\) that a single alarm will detect a burglar. (a) Quota Problem How many such alarms should be used for \(99 \%\) certainty that a burglar trying to enter will be detected by at least one alarm? (b) Suppose the bank installs nine alarms. What is the expected number of alarms that will detect a burglar?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.