Binomial Distribution in Business Statistics: Definition, Formula, Properties, and Applications

This document provides a comprehensive overview of the Binomial Distribution, its properties, formulas, and practical applications within business statistics.

12.1 Formula of Binomial Distribution

The Binomial Distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant for each trial.

The probability mass function (PMF) of a binomial distribution is given by:

$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

Where:

• $P(X=k)$: The probability of getting exactly $k$ successes.
• $n$: The total number of trials.
• $k$: The number of successful outcomes.
• $p$: The probability of success on a single trial.
• $(1-p)$: The probability of failure on a single trial.
• $\binom{n}{k}$: The binomial coefficient, read as "n choose k," which calculates the number of ways to choose $k$ successes from $n$ trials. It is calculated as $\frac{n!}{k!(n-k)!}$.

12.2 Properties of Binomial Distribution

The Binomial Distribution is characterized by the following properties:

1. Fixed Number of Trials ($n$): The experiment consists of a predetermined number of trials.
2. Independent Trials: Each trial is independent of the others. The outcome of one trial does not affect the outcome of any other trial.
3. Two Possible Outcomes: Each trial results in one of two mutually exclusive outcomes, typically referred to as "success" and "failure."
4. Constant Probability of Success ($p$): The probability of success ($p$) is the same for every trial. The probability of failure is therefore $(1-p)$.
5. Discrete Probability Distribution: The random variable (number of successes) can only take on a finite number of values (from 0 to $n$).

12.3 Negative Binomial Distribution

While the binomial distribution focuses on the number of successes in a fixed number of trials, the Negative Binomial Distribution describes the probability of the number of trials required to achieve a fixed number of successes.

For instance, if we want to know the probability of needing 10 trials to achieve 3 successes, we would use the Negative Binomial Distribution. This is distinct from the Binomial Distribution, which would ask for the probability of getting, say, 3 successes in exactly 10 trials.

12.4 Mean and Variance of Binomial Distribution

For a binomial distribution with parameters $n$ (number of trials) and $p$ (probability of success), the mean and variance are:

• Mean (Expected Value): $$\mu = E(X) = np$$ The mean represents the average number of successes expected in $n$ trials.

• Variance: $$\sigma^2 = Var(X) = np(1-p)$$ The variance measures the spread or dispersion of the distribution around its mean.

• Standard Deviation: $$\sigma = \sqrt{np(1-p)}$$ The standard deviation provides a measure of the typical deviation from the mean.

12.5 Shape of Binomial Distribution

The shape of the binomial distribution is influenced by the values of $n$ and $p$:

• When $p < 0.5$: The distribution is skewed to the right (positively skewed). The peak of the distribution is to the left.
• When $p > 0.5$: The distribution is skewed to the left (negatively skewed). The peak of the distribution is to the right.
• When $p = 0.5$: The distribution is symmetric. For large $n$, it approximates a normal distribution.
• As $n$ increases (with $p$ fixed): The distribution becomes more bell-shaped and approaches the normal distribution, especially when $np \ge 5$ and $n(1-p) \ge 5$ (a common rule of thumb).

12.6 Solved Examples of Binomial Distribution

Example 1: Coin Toss Suppose a fair coin is tossed 10 times. What is the probability of getting exactly 6 heads?

• Number of trials ($n$) = 10
• Probability of success (getting a head, $p$) = 0.5
• Number of successes ($k$) = 6

Using the binomial formula: $$P(X=6) = \binom{10}{6} (0.5)^6 (1-0.5)^{10-6}$$ $$P(X=6) = \binom{10}{6} (0.5)^6 (0.5)^4$$ $$P(X=6) = \binom{10}{6} (0.5)^{10}$$

First, calculate the binomial coefficient: $$\binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6!4!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 10 \times 3 \times 7 = 210$$

Now, calculate the probability: $$P(X=6) = 210 \times (0.5)^{10} = 210 \times \frac{1}{1024} \approx 0.2051$$

So, the probability of getting exactly 6 heads in 10 tosses of a fair coin is approximately 0.2051 or 20.51%.

Example 2: Quality Control A manufacturer produces light bulbs, and historical data shows that 2% of the bulbs are defective. If a sample of 50 bulbs is randomly selected, what is the probability that exactly 3 bulbs are defective?

• Number of trials ($n$) = 50
• Probability of success (a bulb being defective, $p$) = 0.02
• Number of successes ($k$) = 3

Using the binomial formula: $$P(X=3) = \binom{50}{3} (0.02)^3 (1-0.02)^{50-3}$$ $$P(X=3) = \binom{50}{3} (0.02)^3 (0.98)^{47}$$

Calculate the binomial coefficient: $$\binom{50}{3} = \frac{50!}{3!(50-3)!} = \frac{50!}{3!47!} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1} = 50 \times 49 \times 8 = 19600$$

Now, calculate the probability: $$P(X=3) = 19600 \times (0.02)^3 \times (0.98)^{47}$$ $$P(X=3) = 19600 \times 0.000008 \times (0.98)^{47}$$

Using a calculator for $(0.98)^{47} \approx 0.3865$: $$P(X=3) \approx 19600 \times 0.000008 \times 0.3865 \approx 0.0604$$

The probability of finding exactly 3 defective bulbs in a sample of 50 is approximately 0.0604 or 6.04%.

12.7 Uses of Binomial Distribution in Business Statistics

The Binomial Distribution is widely applicable in various business contexts:

• Quality Control: Determining the probability of finding a certain number of defective products in a sample.
• Marketing: Estimating the probability of a certain number of customers responding to a campaign or purchasing a product.
• Finance: Modeling the number of profitable trades in a series of independent trades.
• Human Resources: Calculating the probability of a certain number of employees meeting a performance benchmark.
• Operations Management: Analyzing the success rate of service calls or the number of successful equipment operations.

12.8 Real-Life Scenarios of Binomial Distribution

• Survey Responses: The number of people in a sample who answer "yes" to a survey question, assuming each person's response is independent and the probability of answering "yes" is constant.
• Product Testing: Testing a batch of items where each item is either acceptable or defective. The number of acceptable items in a sample follows a binomial distribution.
• Manufacturing Yield: The number of functional units produced from a production line, where each unit has a probability of being functional.
• Investment Decisions: The number of successful investment opportunities identified from a list of potential ventures, where each venture has a probability of being successful.
• Customer Feedback: The number of customers who provide positive feedback out of a surveyed group, assuming each customer's feedback is independent.

12.9 Difference Between Binomial Distribution and Normal Distribution

While distinct, the Normal Distribution serves as a valuable approximation for the Binomial Distribution when the number of trials ($n$) is large, making calculations simpler and allowing for the use of normal distribution tables.