Question

A sporting goods store has a special sale on three brands of tennis balls - call them D, P, and W. Because the sale price is so low, only one can of balls will be sold to each customer. If \(40 \%\) of all customers buy Brand \(\mathrm{W}\), \(35 \%\) buy Brand \(\mathrm{P}\), and \(25 \%\) buy Brand \(\mathrm{D}\) and if \(x\) is the number among three randomly selected customers who buy Brand \(\mathrm{W}\), what is the probability distribution of \(x\) ?

Short answer

The probability distribution of \(x\) can be calculated by following these steps: Set up the problem, define the random variable, apply the binomial distribution formula, and calculate the probability for each possible outcome. The precise distribution will depend on the calculations made in the step 3 and 4.

Step-by-step solution

Identify the scenario

You first need to understand the scenario. The sporting goods store has a special sale on three brands of tennis balls - D, P, and W. Each customer can only buy one can of balls, and the percentages of customers buying each brand are given: 40% buy brand W, 35% buy brand P, and 25% buy brand D.

Define the random variable

Next, you need to define the random variable. Here, \(x\) is the number in 3 randomly selected customers who buy Brand W. Thus, the possible values of \(x\) can be 0, 1, 2, or 3.

Apply the binomial distribution formula

You need to apply the binomial distribution formula, which is \[P(x)={C(n, x) * p^{x} * (1-p)^{n-x}}\] where \(P(x)\) is the probability of \(x\) successes in \(n\) trials, \(C(n, x)\) is the combination of \(n\) items taken \(x\) at a time, \(p\) is the probability of success on a single trial, and \(x\) is the number of successes. Here, \(n=3\), \(p=0.4\), and \(x\) can be 0, 1, 2, or 3.

Calculate the probability for each outcome

Now, plug the values into the formula and calculate the probability for each possible outcome. You will get the probability distribution of \(x\) as a result.

Key concepts

Binomial Distribution

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.

Random Variable

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.

Probability Formula

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.