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Chapter 5: Problem 13

Fundraiser: Hiking Club 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 \(\$ 1\) 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 \(\$ 35\). 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) Interpretation 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
Probabilities: \(\frac{15}{719}\) to win, \(1 - \frac{15}{719}\) not win. Expected earnings: \(\approx \$0.73\). Contribution: \(\approx \$14.27\).

Step by step solution

01

Calculate Total Cookies Sold

First, understand how the game works. In this case, the hiking club sold 719 cookies, each with a different number. Therefore, there are 719 different numbers in the game.
02

Determine Lisa's Winning Probability

Lisa bought 15 cookies, out of a total of 719.To find the probability that Lisa will win, use the formula for probability:\[ P(\text{winning}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{15}{719} \]
03

Calculate Probability Lisa Will Not Win

To find the probability that Lisa will not win, subtract the probability of winning from 1 (since probabilities must sum to 1):\[ P(\text{not winning}) = 1 - P(\text{winning}) = 1 - \frac{15}{719} \]
04

Calculate Expected Earnings

Lisa's expected earnings are calculated by multiplying the probability of winning by the value of the prize (dinner for two, valued at $35):\[ E = 35 \times \frac{15}{719} \]
05

Calculate Lisa's Contribution to the Club

Lisa effectively contributes the amount she paid for the cookies minus her expected earnings:\[ \text{Contribution} = 15 - E \] where E is her expected earnings from Step 4.

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Key Concepts

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

Expected Value
Expected value is a fundamental concept in probability and statistics. It refers to the long-term average of a random variable over a large number of experiments or trials.

In Lisa's case, to find her expected earnings from the fundraiser, we multiply the probability of her winning by the value of the prize. Since the value of the dinner for two is \(\\(35\), we calculate:
  • Probability of winning: \(\frac{15}{719}\)
  • Value of prize: \(\\)35\)
  • Expected earnings: \(E = 35 \times \frac{15}{719}\)
This gives us the expected monetary gain she might receive from purchasing 15 cookies. Expected value provides insight into the average result expected over numerous repetitions of the event.
Combinatorics
Combinatorics is the area of mathematics focused on counting, arrangement, and combination of elements in sets. It's crucial for understanding probability because it helps calculate the number of possible outcomes.

In Lisa's scenario, we consider the cookies, each with unique numbers, akin to tickets in a raffle. Combinatorically speaking, each cookie represents a possible outcome in the drawing.
  • Total outcomes (total cookies): 719
  • Favorable outcomes (cookies Lisa bought): 15
Understanding how to count and arrange the possibilities can help compute probabilities and assess the odds of specific events occurring.
Fundraising Math
Fundraising can involve various mathematical principles, particularly when calculating contributions and expected returns.

The hiking club's fundraiser involves selling fortune cookies, generating funds for new equipment with the prize incentive for participants.
  • Selling price per cookie: \(\\(1\)
  • Total cookies sold: 719
  • Lisa's purchase: 15 cookies
  • Prize value: \(\\)35\)
Lisa's effective contribution is determined by subtracting her expected winnings from the amount spent. The balance illustrates how fundraisers leverage probability and incentives to encourage donations while keeping participants engaged.
Probability Calculation
Calculating probabilities involves determining the likelihood of an event occurring, expressed as a fraction of the favorable outcomes over total outcomes.

For Lisa:
  • Total cookies (events): 719
  • Favorable outcomes (Lisa's cookies): 15
Probability Lisa wins: \(P(\text{winning}) = \frac{15}{719}\)

To find the probability of not winning, subtract the winning probability from 1, as the sum of all probabilities for possible events equals 1.
  • \(P(\text{not winning}) = 1 - \frac{15}{719}\)
Probability calculations provide a means to quantitatively analyze the chance of an event occurring, critical in decision-making processes.
Statistical Analysis
Statistical analysis is the process of collecting and interpreting data to uncover patterns and trends. It often involves using probability theory and other mathematical techniques.

In the context of the cookie fundraiser, statistical analysis can help the hiking club make informed decisions about future fundraisers. By evaluating sales performance, expected earnings, and participant engagement:
  • Analyze the effectiveness of the prize incentive
  • Predict future fundraising success
  • Adjust strategies for greater profit and participation
Statistical analysis turns raw data into actionable insights, allowing organizations to refine processes and enhance outcomes efficiently.

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