Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

Consider \(n\) independent trials of an experiment in which each trial has two possible outcomes, success or failure. The probability of a success on each trial is \(p\) and the probability of a failure is \(q=1-p .\) In this context, the term \(_{n} C_{k} p^{k} q^{n-k}\) in the expansion of \((p+q)^{n}\) gives the probability of \(k\) successes in the \(n\) trials of the experiment. The probability of a sales representative making a sale to any one customer is \(\frac{1}{3} .\) The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term $$_{8} C_{4}\left(\frac{1}{3}\right)^{4}\left(\frac{2}{3}\right)^{4}$$ in the expansion of \(\left(\frac{1}{3}+\frac{2}{3}\right)^{8}\).

Step-by-Step Solution

Answer

The calculation of \(_8C_4\left(\frac{1}{3}\right)^4\left(\frac{2}{3}\right)^4\) yields the probability of the sales representative making 4 sales in a day.

Step 1: Identify the variables

In this case, the number of trials (n) is 8, the number of successes (k) we are looking for is 4, the probability of success (p) is \(\frac{1}{3}\), and the probability of failure (q) is \(\frac{2}{3}\).

Step 2: Apply the binomial probability formula

To find the probability of making 4 sales (successes), we apply the binomial probability formula which is \(_nC_k p^k q^{n-k}\). Plugging in the identified variables, we get \(_8C_4\left(\frac{1}{3}\right)^4\left(\frac{2}{3}\right)^4\).

Step 3: Calculate the combination and probabilities

Next, calculate \(_8C_4\) which is the combination of 8 items taken 4 at a time. Then, compute \(\left(\frac{1}{3}\right)^4 \text{ and } \left(\frac{2}{3}\right)^4\)

Step 4: Complete the calculation

After calculating the combinations and the probabilities, multiply the results together to find the final probability.

Key Concepts

Probability Theory

Probability Theory is a branch of mathematics that deals with the likelihood of different outcomes occurring. It provides a way to quantify uncertainty and model situations where chance is involved. In the context of our problem, we're dealing with two possible outcomes in each trial - success or failure. And these outcomes help us understand real-life situations, like predicting sales or occurrences in various settings.

In probability theory, we're often interested in calculating the chance of multiple events occurring together. The problem with the sales representative uses probability theory to determine the chance of making exactly four sales out of eight contacts. This scenario is an example of a binomial experiment, where each trial is independent, and the probability of success remains constant.

Key concepts in this area include:

• Random Variables - which represent outcomes' possible values.
• Probability Distributions - functions that provide the probabilities of different outcomes.
• Expected Value - the average outcome we expect over many trials.

Understanding these concepts can help you solve various problems and make predictions effectively.

Binomial Coefficient

The Binomial Coefficient is a crucial concept when working with binomial distributions and probability calculations. It is denoted as \(_nC_k\), and it tells us how many ways we can select \(k\) successes in \(n\) independent trials. With the sales representative problem, we are interested in calculating \(_8C_4\) to find the number of ways in which exactly 4 sales (successes) can occur out of 8 attempts.

The formula for the binomial coefficient is:
\[ _nC_k = \frac{n!}{k!(n-k)!} \]
This formula helps us compute combinations, where \(n!\) represents the factorial of \(n\), which is the product of all positive integers up to \(n\). Factorials can be a bit tricky, but remember that \(0! = 1\).

Here are important points about the binomial coefficient:

• Combines with probability measures to evaluate outcomes.
• Always a non-negative integer.
• Central to the binomial probability formula.

Using the binomial coefficient, you get a sense of the "selection" aspect of the trials, which is different from simply "arranging" them.

Independent Trials

The concept of Independent Trials is fundamental to understanding binomial distributions. Independent trials mean that the outcome of one trial does not affect the outcome of another. In the scenario of the sales representative, each customer contact is treated as an independent trial.

In practical terms, independent trials imply:

• The probability of success \(p\) remains constant across trials.
• Each trial result is unaffected by previous trials.

These trials are a basic requirement for applying the binomial distribution. If the trials were not independent, the calculation using binomial probabilities wouldn't be accurate.

To picture this, think about flipping a coin multiple times. Each flip is independent of the others, and the probability of landing heads or tails is constant throughout trials. In our sales case, the same principle applies to ensure each day of sales is analyzed with fresh probability metrics. Understanding this idea helps in setting up and solving binomial problems efficiently.