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Chapter 5: Problem 15

The college hiking club is having a fundraiser to buy new equipment for fall and winter outings. The club is selling Chinese fortune cookies at a price of 1 dollar per cookie. Each cookie contains a piece of paper with a different number written on it. A random drawing will determine which number is the winner of a dinner for two at a local Chinese restaurant. The dinner is valued at 35 dollar. since the fortune cookies were donated to the club, we can ignore the cost of the cookies. The club sold 719 cookies before the drawing. (a) Lisa bought 15 cookies. What is the probability she will win the dinner for two? What is the probability she will not win? (b) Lisa's expected earnings can be found by multiplying the value of the dinner by the probability that she will win. What are Lisa's expected earnings? How much did she effectively contribute to the hiking club?

Short Answer

Expert verified
Probability Lisa wins is \(\frac{15}{719}\) and doesn't is \(\frac{704}{719}\). Expected earnings are about 0.73 dollars, and her effective contribution to the club is about 14.27 dollars.

Step by step solution

01

Determine Total Possible Outcomes

The total number of possible outcomes in this scenario is equal to the total number of cookies sold, which is 719. Each cookie sold represents one outcome where one winner will be randomly drawn.
02

Calculate Probability of Winning

Lisa bought 15 cookies, so she has 15 chances out of 719 to win the dinner. The probability that Lisa wins is given by \( P(\text{win}) = \frac{15}{719} \).
03

Probability of Not Winning

The probability that Lisa does not win is the complement of winning. Thus, \( P(\text{not win}) = 1 - P(\text{win}) = 1 - \frac{15}{719} = \frac{704}{719} \).
04

Compute Lisa's Expected Earnings

Lisa's expected earnings from the raffle can be calculated by multiplying the value of the dinner ($35) by the probability of winning: \( E = 35 \times \frac{15}{719} \).
05

Calculate Lisa's Contribution

Lisa paid \(1 per cookie and bought 15 cookies, spending a total of \)15. Her expected earnings are subtracted from this amount to find her effective contribution: \( \text{Contribution} = 15 - E \).

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Key Concepts

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

Expected Value
The expectation, or expected value, is a central concept in probability that predicts the average outcome when an experiment is repeated many times. For a probability event, the expected value can be thought of as the sum of all possible outcomes, each multiplied by their respective probabilities.
To understand this better, let's consider Lisa's situation in the raffle. She wants to know what she can expect to "earn" on average when she buys cookies. The expected value is found by multiplying the value of the prize, in this case a dinner worth $35, by her probability of winning.
So, Lisa's expected earnings are calculated by the formula:- Expected Value: 35 times the probability of her winning.
Using the formula, Lisa's expected earnings from her purchase of 15 cookies in a raffle where 719 total cookies were sold is:\[E = 35 \times \frac{15}{719}\]This operation will give a value that tells Lisa the average "payoff" of her participation, essentially giving her an insight into the risk and reward of her decision to join the raffle.
Complementary Probability
In probability, calculating complementary probabilities is a handy way to determine how likely an event is to NOT happen. The complement rule is simple: the probability of an event happening plus the probability of it not happening totals 1.
For Lisa, who is trying to figure out the odds of not winning in the raffle, the complementary probability is calculated based on the probability of winning.
Here, she bought 15 out of 719 cookies. So:- Probability of winning: \( \frac{15}{719} \)- Complementary Probability (Probability of not winning):\[P(\text{not win}) = 1 - \frac{15}{719} \]In Lisa's case, the complementary probability offers a clear understanding of her likelihood of not winning, which can be helpful for setting expectations and making informed decisions about participating in similar games of chance.
Raffle
A raffle is a common fundraising method where participants buy tickets or entries for a chance to win a prize. The allure of a raffle lies in the mix of chance and reward, which can make fundraising efforts successful by drawing in many participants. Each entry, whether a ticket or a cookie as in Lisa's situation, represents an equal chance of winning the designated prize.

In the case of the hiking club fundraiser, they employed a raffle using Chinese fortune cookies. Each cookie is purchased for $1, and inside one cookie is the winning number for a dinner valued at $35. Selling these cookies allows raising funds efficiently. Once the cookies are all sold, one is drawn at random to determine the winner.
  • This method of fundraising can boost participation because it offers a clear prize.
  • The nonprofit aspect (assuming cookie costs are covered) enhances fundraising potential.
Raffles capitalize on people's chance-driven excitement, often creating a win-win by generating funds for projects while offering participants a chance at something beyond their purchase.
Fundraising
Fundraising is the practice of gathering voluntary financial contributions to support a cause. For the college hiking club, the need to gather funds for new equipment prompted them to engage in a fundraising activity by selling fortune cookies.
In particular, fundraisers like this rely on community engagement and the attractiveness of potential rewards to encourage participation. Here are a few key elements of effective fundraising:
  • Clarity of Purpose: Explain why funds are needed; in this case, for hiking equipment.
  • Incentive: Providing a prize, e.g., a dinner valued at $35, increases interest.
  • Cost Management: Since the cookies were donated, all sales go directly to fundraising efforts.
Lisa's experience highlights how a relatively small personal investment can contribute to a larger community goal, with her purchase both supporting the club and potentially leading to a personal reward, making it attractive to join and support.

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Most popular questions from this chapter

In Hawaii, January is a favorite month for surfing since \(60 \%\) of the days have a surf of at least 6 feet (Reference: Hawaii Data Book, Robert C. Schmitt). You work day shifts in a Honolulu hospital emergency room. At the beginning of each month you select your days off, and you pick 7 days at random in January to go surfing. Let \(r\) be the number of days the surf is at least 6 feet. (a) Make a histogram of the probability distribution of \(r .\) (b) What is the probability of getting 5 or more days when the surf is at least 6 feet? (c) What is the probability of getting fewer than 3 days when the surf is at least 6 feet? (d) What is the expected number of days when the surf will be at least 6 feet? (e) What is the standard deviation of the \(r\) -probability distribution? (f) Interpretation Can you be fairly confident that the surf will be at least 6 feet high on one of your days off? Explain.

Expand Your Knowledge: Multinomial Probability Distribution Consider a multinomial experiment. This means the following: 1\. The trials are independent and repeated under identical conditions. 2\. The outcomes of each trial falls into exactly one of \(k \geq 2\) categories. 3\. The probability that the outcomes of a single trial will fall into ith category is \(p_{i}\) (where \(i=1,2 \ldots, k\) ) and remains the same for each trial. Furthermore, \(p_{1}+p_{2}+\ldots+p_{k}=1\) 4\. Let \(r_{i}\) be a random variable that represents the number of trials in which the outcomes falls into category \(i\). If you have \(n\) trials, then \(r_{1}+r_{2}+\ldots\) \(+r_{k}=n .\) The multinational probability distribution is then $$P\left(r_{1}, r_{2}, \cdots r_{k}\right)=\frac{n !}{r_{1} ! r_{2} ! \cdots r_{2} !} p_{1}^{r_{1}} p_{2}^{(2)} \cdots p_{k}^{r_{2}}$$ How are the multinomial distribution and the binomial distribution related? For the special case \(k=2,\) we use the notation \(r_{1}=r, r_{2}=n-r, p_{1}=p\) and \(p_{2}=q .\) In this special case, the multinomial distribution becomes the binomial distribution. The city of Boulder, Colorado is having an election to determine the establishment of a new municipal electrical power plant. The new plant would emphasize renewable energy (e.g., wind, solar, geothermal). A recent large survey of Boulder voters showed \(50 \%\) favor the new plant, \(30 \%\) oppose it, and \(20 \%\) are undecided. Let \(p_{1}=0.5, p_{2}=0.3,\) and \(p_{3}=0.2 .\) Suppose a random sample of \(n=6\) Boulder voters is taken. What is the probability that (a) \(r_{1}=3\) favor, \(r_{2}=2\) oppose, and \(r_{3}=1\) are undecided regarding the new power plant? (b) \(r_{1}=4\) favor, \(r_{2}=2\) oppose, and \(r_{3}=0\) are undecided regarding the new power plant?

The Honolulu Advertiser stated that in Honolulu there was an average of 661 burglaries per 100,000 households in a given year. In the Kohola Drive neighborhood there are 316 homes. Let \(r=\) number of these homes that will be burglarized in a year. (a) Explain why the Poisson approximation to the binomial would be a good choice for the random variable \(r\). What is \(n\) ? What is \(p ?\) What is \(\lambda\) to the nearest tenth? (b) What is the probability that there will be no burglaries this year in the Kohola Drive neighborhood? (c) What is the probability that there will be no more than one burglary in the Kohola Drive neighborhood? (d) What is the probability that there will be two or more burglaries in the Kohola Drive neighborhood?

Sara is a 60 -year-old Anglo female in reasonably good health. She wants to take out a 50,000 dollar term (i.e., 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). $$\begin{array}{|l|lcccc|} \hline x=\text { age } & 60 & 61 & 62 & 63 & 64 \\\ \hline P( \text { death at this age) } & 0.00756 & 0.00825 & 0.00896 & 0.00965 & 0.01035 \\ \hline \end{array}$$ 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 dollar 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) If Big Rock Insurance wants to make a profit of 700 dollar above the expected total cost paid out for Sara's death, how much should it charge for the policy? (d) If Big Rock Insurance Company charges 5000 dollar for the policy, how much profit does the company expect to make?

Suppose we have a binomial experiment, and the probability of success on a single trial is 0.02. If there are 150 trials, is it appropriate to use the Poisson distribution to approximate the probability of three successes? Explain.
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