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