The **aim of this question** is to understand the key concept of a **random variable** using the **coin toss experiment** which is the most basic **binomial (experiment with two possible outcomes) experiment** performed in probability theory.

A **random variable** is nothing but **a mathematical formula** used to describe the **outcome of statistical experiments**. For example, $X$ is a random variable defined as the difference of head and tail outcomes out of $n$ experiments in this question.

The **concept of random variables is essential** for understanding the further key concepts of process probability and its functions.

## Expert Answer

**Let:**

\[ \text{ total number of coin tosses } \ = \ n \]

**And:**

\[ \text{ number of tails } \ = \ t \]

Then, the **no. of heads** can be found using following formula:

\[ \text{ number of heads } \ = \ h \ = \ n \ – \ t \]

Since $X$ is defined as the **difference of total number of heads and tails**, it can be calculated using following formula:

\[ X \ = h \ – \ t \ = \ ( \ n \ – \ t \ ) \ – \ t \ = \ h \ – \ t \ – \ t \ = \ h \ – \ 2t \ \]

Thus **possible values of $X$** can be written in mathematical form as:

\[ X \ = \ \bigg \{ \ n \ – \ 2t \ \bigg | \ t \ = \ \{ \ 0, \ 1, \ 2, \ , ……, \ n \ \} \ \bigg \} \]

## Numerical Result

\[ \text{ Possible values of } X \ = \ \bigg \{ \ n \ – \ 2t \ \bigg | \ t \ = \ \{ \ 0, \ 1, \ 2, \ , ……, \ n \ \} \ \bigg \} \]

## Example

**A coin is tossed 100 times and tail came up in 45 experiments. Find the value of $X$.**

**For this case:**

\[ n \ = \ 100 \]

\[ t \ = \ 45 \]

**Hence:**

\[ h \ = \ 100 \ – \ 45 \ = \ 55 \]

**$X$** can be calculated using following formula:

**\[ X \ = 55 \ – \ 45 \ = \ 10 \]**

Which is the value of $X$ when $45$ tails show up in $100$ coin tosses