Introduction

The binomial distribution is a fundamental concept in probability theory that models the outcomes of a series of independent and identical trials. These trials are characterized by two possible outcomes: success or failure. The binomial distribution provides insights into the probabilities of obtaining a specific number of successes in a fixed number of trials. This chapter explores the properties, formula, and real-world applications of the binomial distribution.

Properties of the Binomial Distribution

The binomial distribution is characterized by the following properties:

• Each trial has only two possible outcomes: success or failure.
• The probability of success remains constant across all trials.
• Trials are independent of each other.
• The distribution is discrete, as it deals with a count of successes.

Binomial Probability Formula

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

$$P(X=k)=\left(\begin{array}{l} n \\ k \end{array}\right) \cdot p^k \cdot(1-p)^{n-k}$$

Where:

• $n$ is the number of trials
• $k$ is the number of successes
• $p$ is the probability of success in a single trial
• $1-p$ is the probability of failure in a single trial
• $\left(\begin{array}{l}n \\ k\end{array}\right)$ represents the binomial coefficient, also known as “n choose k,” and is calculated as $\frac{n !}{k ! \cdot(n-k) !}$

Example: Suppose a fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

• $n=5$ (number of trials)
• $k=3$ (number of successes)
• $p=0.5$ (probability of heads) Using the binomial formula:

$P(X=3)=\left(\begin{array}{l} 5 \\ 3 \end{array}\right) \cdot(0.5)^3 \cdot(0.5)^{5-3}$

$P(X=3)=10 \cdot 0.125 \cdot 0.125=0.3125$