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Chapter 15: Problem 24

Justine works for an organization committed to raising money for Alzheimer's research. From past experience, the organization knows that about \(20 \%\) of all potential donors will agree to give something if contacted by phone. They also know that of all people donating, about \(5 \%\) will give \(\$ 100\) or more. On average, how many potential donors will she have to contact until she gets her first \(\$ 100\) donor?

Short Answer

Expert verified
On average, Justine will have to contact approximately 100 potential donors until she gets her first $100 donor.

Step by step solution

01

Determine the Probability of Giving $100

First, we need to determine the overall chance that a given potential donor will donate $100. The chance that someone will donate is 20%, or 0.20. Given they have agreed to donate, the chance they will donate $100 or more is 5%, or 0.05. Since these are independent events, the combined probability is their product: \(0.20 脳 0.05 = 0.01\), or 1%.
02

Calculate Average Number of Contacts

Assuming that the donors are contacted randomly and each donor independently decides whether to donate, this is a geometric distribution problem. The expected value (average or mean) of a geometrically distributed random variable is \(1/p\) where \(p\) is the probability of the 'success' event. In our case, the 'success' event is that the contacted individual donates $100 or more, which occurs with probability 1%. Thus, the expected number of potential donors contacted until a success is \(1/0.01 = 100\).
03

Interpret the Result

The calculated expected value means that on average, Justine will need to contact about 100 potential donors to get one who will donate $100 or more. It is important to note that the number can vary due to randomness, and she might need to contact more or fewer donors in practice to get a $100 donor.

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Key Concepts

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

Probability Theory
Probability theory is the branch of mathematics that deals with quantifying the likelihood of events. It's essential for understanding various random processes which underpin a wide variety of disciplines from physics to finance, and, as in this example, philanthropy efforts.

Probability is measured on a scale from 0 to 1, where 0 implies an impossible event and 1 represents an event that is certain to happen. Intermediate values are assigned to events based on their likelihood. For instance, Justine knows from her organization's experience that approximately 20% (\(0.20\) when expressed as a probability) of potential donors will agree to donate鈥攁 useful figure when planning outreach strategies.
Expected Value
The concept of expected value is pivotal in understanding the long-term average outcome of random events. It is essentially a weighted average of all possible outcomes, where the weights are their respective probabilities. To put it simply, it tells us what to expect on average over many trials of the same experiment.

In the case of Justine and her fundraising efforts, the expected value indicates the average number of potential donors needed to contact before securing a donation of \(100 or more. Calculating this requires understanding the geometric distribution, which models the number of trials needed to achieve a first success. Using the formula for the expected value of a geometrically distributed variable, \(1/p\), with \(p\) being the probability of a donor contributing \)100, Justine could expect to call 100 potential donors on average to achieve her goal.
Independent Events
  • An independent event is one whose occurrence is not affected by any other events.
  • For example, each potential donor's decision to give \(100 is independent of another's decision.

Understanding this concept is crucial when dealing with probabilities of combined events in the context of these charitable contributions. If two events are independent, as is the case with separate individuals deciding to donate, the probability of both events occurring is the product of their individual probabilities. In the exercise, the independent contribution decisions of potential donors are used to determine the overall probability that someone will donate \)100. This independent probability is 1%, found by multiplying the probability of agreeing to donate (20%) by the probability of donating $100 (5%).

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Most popular questions from this chapter

About \(8 \%\) of males are color-blind. A researcher needs some color-blind subjects for an experiment and begins checking potential subjects. a. On average, how many men should the researcher expect to check to find one who is color-blind? b. What's the probability that she won't find anyone colorblind among the first 4 men she checks? c. What's the probability that the first color-blind man found will be the sixth person checked? d. What's the probability that she finds someone who is color-blind before checking the 10th man?

If you flip a fair coin 100 times, a. Intuitively, how many heads do you expect? b. Use the formula for expected value to verify your intuition.

A manufacturer ships toasters in cartons of 20 . In each carton, they estimate a \(5 \%\) chance that one of the toasters will need to be sent back for minor repairs. What is the probability that in a carton, there will be exactly 3 toasters that need repair?

Do these situations involve Bernoulli trials? Explain. a. You are rolling 5 dice and need to get at least two 6 's to win the game. b. We record the distribution of eye colors found in a group of 500 people. c. A manufacturer recalls a doll because about \(3 \%\) have buttons that are not properly attached. Customers return 37 of these dolls to the local toy store. Is the manufacturer likely to find any dangerous buttons?

The basketball player in Exercise 55 has new sneakers, which he thinks improve his game. Over his past 40 shots, he's made 32 -much better than the \(55 \%\) he usually shoots. Do you think his chances of making a shot really increased? In other words, is making at least 32 of 40 shots really unusual for him? (Do you think it's his sneakers?)
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