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Understanding binomial distribution is crucial for solving probability problems involving discrete events where there are exactly two outcomes, like success and failure. In the scenario provided, we are looking at how many customers out of a random sample of three will choose Brand W tennis balls, given a fixed probability.

A binomial distribution is defined by two parameters: the number of trials (n) and the probability of success in a single trial (p). In this context, a 'trial' refers to a single customer's purchase, and 'success' means the customer buys Brand W. The formula for binomial distribution is expressed as \[ P(x)={C(n, x) * p^{x} * (1-p)^{n-x}} \] where \( C(n, x) \) represents the number of combinations of \(n\) items taken \(x\) at a time.

This formula is handy to calculate the likelihood of any number of successes (customers buying Brand W) across a series of trials (total customers approached). It forms the backbone of many probability questions and helps us understand the likelihood of different outcomes.

In our exercise, the term 'random variable' refers to a numerical value that results from a random phenomenon. The random variable \(x\) represents the number of customers who buy Brand W. Since each customer can only be classified as a 'success' (buying Brand W) or a 'failure' (buying any other brand), \(x\) can only take on the values 0, 1, 2, or 3 in our set of three customers, making it a discrete random variable.

Understanding random variables is vital, as they provide a way to quantify random events. The values of \(x\) in this context are calculated using the probabilities of customer choices, which allows us to build a probability distribution—a list of the probabilities associated with each possible outcome.

The probability formula gives us a way to find the likelihood of an event. In general terms, it can be written as \( P(Event) = \frac{Number \; of \; favorable \; outcomes}{Total \; number \; of \; possible \; outcomes} \). However, in the case of the binomial distribution, we use a more specific formula as highlighted in the previous sections.

For our binomial distribution example, the probability of zero customers choosing Brand W (zero successes) is calculated by plugging \(x=0\) into the formula, and similarly for one, two, and three customers choosing Brand W. Using the formula, the more comprehensive set of solutions for a binomially distributed random variable ensures a deeper understanding of the distribution of outcomes based on the given probabilities of each brand being chosen.