Question

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

Justine will need to contact 100 donors on average to get her first $100 donor.

Step-by-step solution

Understanding the Problem

Justine needs to find out how many phone calls are needed to secure a donation of \\(100 or more. We need to find two probabilities: the probability a contacted donor gives a donation, and within those donors, the probability of donating \\)100 or more.

Probability of Getting a Donor

We are told that 20% of the contacted potential donors agree to donate. Therefore, the probability of one potential donor agreeing to donate when contacted is \( p_1 = 0.20 \).

Probability of Donating $100 or More

Given the donors, the probability that a donor gives \$100 or more is 5%. Therefore, the probability is \( p_2 = 0.05 \).

Combined Probability of $100 Donation

The overall probability of contacting a potential donor and receiving a \$100 donation is the product of the two probabilities: \( p = p_1 \times p_2 = 0.20 \times 0.05 = 0.01 \).

Mean Number of Contacts Using Geometric Distribution

The problem asks for the expected (average) number of potential donors Justine must contact. This is a geometric distribution problem where the probability of success (getting a \$100 donor) is \( p = 0.01 \). The mean (expected value) of a geometric distribution is \( \frac{1}{p} \).

Calculate the Expected Number of Contacts

Using the formula for the mean of a geometric distribution, we calculate \[ \text{Mean} = \frac{1}{p} = \frac{1}{0.01} = 100. \] Therefore, on average, Justine will have to contact 100 donors to get one \$100 donation.

Key concepts

Probability

Probability helps us quantify the likelihood of a particular event happening. In this scenario, we are looking at the chances of potential donors contributing to Alzheimer's research. The organization already has some useful insights about their donor interactions. Let's break it down.

• The probability that a potential donor will agree to donate once contacted is 20%, which translates to a numerical probability of \( p_1 = 0.20 \).
• Among those who donate, there's a 5% chance they'll contribute \(100 or more, which translates to \( p_2 = 0.05 \).

Combining these probabilities helps us understand the overall chance of getting a specific donation. By multiplying the two probabilities, we find the likelihood of a donor giving \)100 or more, calculated as \( p = p_1 \times p_2 = 0.20 \times 0.05 = 0.01 \). Hence, there is a 1% chance that any contacted potential donor will donate $100 or more.

Expected Value

The concept of expected value is very important in probability and statistics because it tells us about the long-term average outcome of a random process. In this task, we want to calculate the expected number of donors Justine needs to contact to get her first \(100 donor using a geometric distribution.
A geometric distribution is used when we look for the number of trials until the first success. Here, success is considered when a donor gives at least \)100. The expected value, or mean, in a geometric distribution is calculated using the formula \( \frac{1}{p} \), where \( p \) is the probability of the event.

• Given the probability of receiving a \(100 donation is \( 0.01 \), we use the formula: \[ \text{Mean} = \frac{1}{0.01} = 100. \]
• This result tells us that Justine needs to contact an average of 100 potential donors to secure her first \)100 donation.

Understanding expected value can help set realistic goals and make informed decisions, focusing efforts where they are most likely to pay off.

Donor Behavior

In charitable fundraising, understanding donor behavior is crucial for optimizing resources and efforts. Organizations can tailor their outreach strategies by analyzing past interactions and understanding donor tendencies.

• In this example, prior data reveals that out of all potential donors, only 20% typically decide to donate when reached by phone. This insight allows Justine and her team to anticipate and segment their outreach efforts effectively.
• Among those who do donate, only 5% are inclined to donate $100 or more. This information is valuable as it helps the organization focus on potential high-value donors without expending resources on non-donors.

By understanding these behavior patterns, the organization can improve the efficiency of its fundraising efforts. They can identify the best contact methods, the right times to connect with potential donors, and customize their messaging to resonate with different donor segments. These strategies can significantly impact overall fundraising outcomes, allowing the organization to achieve its goals faster and more economically.