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 dollars or more. On average, how many potential donors will she have to contact until she gets her first 100 dollars donor?

The main objective of this question is to find the number of calls in order to get a donation of 100 dollars from these calls.

This question uses the concept of  Binomial probability. In binomial distribution, we have two possible outcomes for a trial, which is success or failure.

We are given that $20 %$ of the donors will be donating if they are contacted by someone. About $5 %$ of donors will be donating more than $100$ dollars.

We have to find the number of calls in order to get a donation of 100 dollars from these calls.

So the probability of success is:

$= \space 5 % \space \times \space20%$

$=\space \frac{5}{100} \times \frac{20}{100}$

$= \space \frac{100}{10000}$

$=\space 0.01$

$= \space 1 \space %] Now: \[E(x) \space = \space \frac{1}{p}$

$E(x) \space = \space \frac{1}{0.01}$

$E(x) \space = \space 100$

The number of calls will be $100$ in order to get a donation of $100$ dollars.

Example

Find the number of calls in order to get a donation of $100$ dollars from these calls. The $20 %$, $40 %$, and $60 %$ of the donors will be donating if they are contacted by someone while  $10 %$ donors will be donating more than $100$ dollars.

First, we will solve it for $20 %$.

We are given that $20 %$ of the donors will be donating if they are contacted by someone. About $10 %$ donors will be donating more than $100$ dollars.

We have to find the number of calls in order to get a donation of  $100$ dollars from these calls.

So the probability of success is:

$= \space 10 % \space \times \space20%$

$=\space \frac{10}{100} \times \frac{20}{100}$

$= \space \frac{200}{10000}$

$=\space 0.02$

Now:

$E(x) \space = \space \frac{1}{p}$

$E(x) \space = \space \frac{1}{0.02}$

$E(x) \space = \space 50$

Now solving it for $40 %$.

We are given that $20 %$ of the donors will be donating if they are contacted by someone. About $40 %$ of donors will be donating more than $100$ dollars.

We have to find the number of calls in order to get a donation of 100 dollars from these calls.

So the probability of success is:

$= \space 10 % \space \times \space20%$

$=\space \frac{40}{100} \times \frac{20}{100}$

$= \space \frac{800}{10000}$

$=\space 0.08$

Now:

$E(x) \space = \space \frac{1}{p}$

$E(x) \space = \space \frac{1}{0.08}$

$E(x) \space = \space 12.50$

Now solving it for $60 %$.

We are given that $20 %$ of the donors will be donating if they are contacted by someone. About $60 %$ of donors will be donating more than $100$ dollars.

We have to find the number of calls in order to get the donation of 100 dollars from these calls.

So the probability of success is:

$= \space 10 % \space \times \space20%$

$=\space \frac{60}{100} \times \frac{20}{100}$

$= \space \frac{1200}{10000}$

$=\space 0.12$

Now:

$E(x) \space = \space \frac{1}{p}$

$E(x) \space = \space \frac{1}{0.12}$

$E(x) \space = \space 8.33$