Binomial Distribution

What is it?

The binomial probability distribution is a Statistical Distribution user in Statistics for modelling events that have discrete binomial variables, which can only take two values, positive or negative.

When applied to a series of events, with two possible outcomes, it describes the Probability of getting a certain number of successes in a series of trials.

For example, what is the probability of getting 4 heads in 10 coin tosses? We can easily plot the binomial distribution using SciPy and Python:

import numpy as np
import seaborn as sns
from scipy.stats import binom
 
number_tosses = 10
probability_success = 0.5  # Assuming a fair coin
 
x_values = np.arange(0, n + 1)
 
probabilities = binom.pmf(x_values, n, p)
 
# Plot the binomial distribution
sns.barplot(y=probabilities, x=x_values)
plt.title("Binomial distribution for 10 coin tosses");

---

Bernoulli trials and its assumptions

Each coin toss in the past example can be referred as a Bernoulli trial. In reality, each event of a binomial distribution is a Bernoulli trial, which in itself makes some assumptions to properly model the events.

• There must be only two possible outcomes. If the **success rate is $p$, then the failure rate is $1-p$.

• Each outcome must have a fixed probability of occurring.

• Each trial is completely independent of all others.

Modelling non-independent events

Binomial distribution experiments are made with replacement so each event is independent from one another. For modelling non-independent events, one can use the Hypergeometric distribution.

---

Calculating the probability

Given the number of successes $x$, number of trials $n$, probability of success $p$, and probability of failure $q$, one could use the Probability Mass Function of the binomial distribution to calculate the distribution of a experiment:

$$P(x)=\frac{n!}{(n-x)!x!}{p}^x{q}^{n-x}$$

Because the average always tend to $p$, the mean of the binomial probability distribution is $\overline{x}=np$, and Variance is ${\sigma }^2=npq$.