Binomial Distribution Explained: Definition, Theory, and Applications

🔎 Introduction

The binomial distribution is one of the most fundamental discrete probability distributions in statistics. It describes the probability of obtaining a certain number of successes in a fixed number of independent trials, when each trial has only two outcomes: success (with probability $p$) or failure (with probability $q=1-p$).

This distribution is a building block for more advanced models like the multinomial distribution.

📘 Definition

If $X$ is the number of successes in $n$ independent trials, then:

Figure 1: Binomial distribution probability mass function (PMF).

$$ P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0,1,2,\dots,n $$

Here:

$n$ = number of trials
$p$ = probability of success
$q=1-p$ = probability of failure
$\binom{n}{x}$ = number of ways to choose $x$ successes from $n$ trials

📐 Properties

Mean: $E[X] = np$
Variance: $\mathrm{Var}(X) = np(1-p)$
Mode: $\lfloor (n+1)p \rfloor$
Shape:
- Symmetric when $p=0.5$
- Right-skewed if $p<0 .5="" li="">
- Left-skewed if \(p>0.5$

🌍 Real-Life Applications

Quality Control: Counting defective vs. non-defective products in a batch.
Medicine: Number of patients responding positively to a new drug.
Education: Number of correct answers in a multiple-choice test.
Sports: Number of successful goals/shots in a fixed number of attempts.
Genetics: Inheritance of dominant vs. recessive traits.

📊 Example

Problem: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Solution:

$$ P(X=3) = \binom{5}{3} (0.5)^3 (0.5)^2 $$

$\binom{5}{3} = 10$
$(0.5)^5 = 0.03125$
$P(X=3) = 10 \times 0.03125 = 0.3125$

So, the probability is 0.3125 (31.25%).

Figure 2: Binomial distribution for $n=5, p=0.5$. Probabilities for 0–5 successes.

📝 Key Takeaways

The binomial distribution models binary outcomes (success/failure).
Its PMF: $\binom{n}{x} p^x (1-p)^{n-x}$.
Mean = $np$; Variance = $np(1-p)$.
Applications range from quality control to medicine, education, and sports.
It is the foundation for more general distributions like the multinomial.

👉 Related Posts

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Sunday, December 11, 2011