Difference Between Binomial and Poisson Distribution

	BINOMIAL	POISSON
Number of Trials and Outcomes	Fixed number of trials (n) with exactly two possible outcomes (often denoted as success and failure).Examples: Flipping a coin 5 times (n=5) with heads (success) and tails (failure) as outcomes.	Focuses on events occurring in a fixed interval (like time or space) with no limit on the number of possible events (can be zero, one, two, and so on).Examples: Number of customer arrivals in a store in an hour.
Probability	Deals with the probability of getting a specific number of successes (r) out of the total trials (n). It requires knowing the probability of success (p) beforehand, which remains constant throughout the trials.	Models the probability of observing a certain number of events (k) within the interval.It uses an average rate (λ) of event occurrence, not the probability of individual events.
Mean and Variance	Mean (μ) is n * p (number of trials times probability of success), and variance is n * p * (1 – p).	Both the mean (μ) and variance are equal to the average rate (λ) of event occurrence.

Choosing the Right Distribution:

Use the binomial distribution when you have a fixed number of independent trials with two possible outcomes, and you want to know the probability of a specific number of successes.
Use the Poisson distribution when you’re dealing with events happening independently over a fixed interval, and you’re interested in the probability of a certain number of events occurring within that interval.
Here’s an analogy to illustrate the difference:

Imagine rolling a fair die multiple times (fixed number of trials). The binomial distribution can tell you the probability of getting exactly two sixes (successes) in five rolls (n=5).
Poisson distribution, on the other hand, is helpful if you’re concerned about the number of car accidents at a particular intersection every day (fixed interval).expand_more It can’t predict individual accidents, but it can model the likelihood of having, say, three accidents on a specific day based on the average daily accident rate.

Example

from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns

sns.distplot(random.normal(loc=10, scale=10, size=1000), hist=False, label='normal')
sns.distplot(random.poisson(lam=10, size=1000), hist=False, label='poisson')

plt.show()

Output