On the Internet you can find a number of definitions of a binomial random variable, see Wikipedia, Stat Trek or PennState, among others. None of them seems to me as intuitive as the one provided here. The definition is given in three steps.
Step 1. Everybody knows what is a coin: it takes values and with equal probabilities and . With this information, it is easy to understand what is an unfair coin: it takes value with probability and with probability , where is some number between and . This definition describes what happens when we toss an unfair coin once.
Step 2. Now let us toss the coin times and count the number of successes (getting means success and getting means failure). The random variable that describes the number of successes is called a binomial variable. There is little you can do with this definition; we need to make one more step. Let us denote the binomial variable and let , ..., be the outcomes on the coins. The fundamental fact suggested by the procedure of counting the number of successes is that .
To illustrate this equation, consider the case . There are possible combinations of the outcomes on the two coins: 1) , 2) , 3) and 4) . Plug the coin values in the equation , and you will see that in each case the equation is true.
Regarding our experiment of tossing the coin times, two remarks are in order: 1) obviously, the outcomes are independent and 2) the coins are identically distributed in the sense that the probability does not change throughout the experiment.
Step 3. It is possible to give different (and equivalent) definitions for the same thing. The one that takes the bull by the horns and can be directly applied is called a working definition. For the binomial random variable, the working definition is this: it is a sum of independent identically distributed unfair coins. That is, you write and then specify that the coins are independent and have the same .
Every Economics student knows that the market demand is equal to the sum of individual demands: . The definition of the binomial variable is a perfect analog of this fact. Sums of random variables are omnipresent in Statistics and Theory of Probabilities. By omitting the working definition of the binomial variable, elementary Statistics textbooks, including AP Stats and Business Stats, miss the essence of Statistics.