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**:

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

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

\[E(x) \space = \space 8.33 \]