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.
Expert Answer
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:
\[E(x) \space = \space \frac{1}{p} \]
\[E(x) \space = \space \frac{1}{0.01} \]
\[E(x) \space = \space 100 \]
Numerical Answer
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:
\[E(x) \space = \space \frac{1}{p} \]
\[E(x) \space = \space \frac{1}{0.12} \]
\[E(x) \space = \space 8.33 \]