Probability Distributions Explained: Binomial, Poisson, Normal

Think of the admissions desk at a training institute in Indore. On a normal weekday it gets about 4 phone enquiries an hour. Some hours bring 2 calls, some bring 7, and once in a while an hour passes with none.

The manager wants to know how many people to staff. "Four on average" is not enough. She needs to know how often 8 or more calls arrive at once, because that is when callers hang up.

What she needs is the full picture: the chance of 0 calls, 1 call, 2 calls and so on. That picture is a probability distribution.

What is a probability distribution?

Map the call desk to the terms:

• "Number of calls in one hour" is a random variable: a quantity whose value depends on chance.
• The values it can take (0, 1, 2, 3, …) form its support.
• The table of "value → probability" is its distribution.

A probability distribution is a function that gives the probability of every possible value of a random variable, and all those probabilities add up to 1.

The probability basics tutorial dealt with single events, such as "this order is late". A distribution describes every outcome at once, so it answers any question about the variable: its average, its spread and the chance of extremes.

Discrete vs continuous random variables

The first question to ask about any column is whether it is counted or measured.

The last row surprises many beginners. For a continuous variable, the chance that a delivery takes exactly 30.000000 minutes is zero. You can only ask about intervals, and the probability is the area under the density curve over that interval.

A PMF puts probability on separate bars; a PDF spreads it as area under a curve.

Both kinds share one tool, the CDF (cumulative distribution function). CDF(x) is the probability that the variable is at most x. "What is the chance of 5 or fewer calls?" is a CDF question.

Discrete distributions: Bernoulli, binomial and Poisson

Bernoulli: one yes-or-no trial

A Bernoulli variable takes the value 1 with probability p and 0 with probability 1 − p. One visitor either signs up for a demo or not. It is the building block for the next two.

Binomial: successes in n trials

Run n independent Bernoulli trials with the same p and count the successes. That count follows a binomial distribution, written Binomial(n, p).

Say 20% of people who call the desk book a demo class. Out of 10 callers today, how many will book? The formula is P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) counts the ways to choose which k callers book. Python's math.comb computes C(n, k) directly.

from math import comb

n, p = 10, 0.20
for k in range(0, 7):
    prob = comb(n, k) * p**k * (1 - p)**(n - k)
    print(f"P({k} bookings) = {prob:.3f}  " + "#" * round(prob * 60))

mean = n * p
var = n * p * (1 - p)
print(f"\nMean = {mean}, variance = {var:.2f}")
print(f"P(4 or more) = {1 - sum(comb(n,k)*p**k*(1-p)**(n-k) for k in range(4)):.3f}")

Two bookings is the single most likely result, and four or more bookings happens on roughly 12% of days. Mean n × p and variance n × p × (1 − p) are worth remembering.

Poisson: events in a fixed window

When you count events in a time or space window and there is no fixed number of trials, use the Poisson distribution. It has one parameter, λ (lambda), the average count per window. Calls per hour, website errors per day and potholes per kilometre of road are classic Poisson data.

For the call desk with λ = 4, P(X = k) = e^(−λ) × λᵏ / k!. The manager's question was the chance of 8 or more calls in an hour.

from math import exp, factorial

lam = 4
pmf = [exp(-lam) * lam**k / factorial(k) for k in range(20)]

for k in range(0, 10):
    print(f"P({k} calls) = {pmf[k]:.3f}")

p_8_plus = 1 - sum(pmf[:8])
print(f"\nP(8 or more calls in an hour) = {p_8_plus:.3f}")
print(f"Busy hours in a 9-hour day, on average: {9 * p_8_plus:.2f}")

About 5% of hours bring 8 or more calls. In a 9-hour day, that is roughly one rush hour every two days, a number she can actually plan around.

Continuous distributions: uniform, normal and exponential

Uniform

In a uniform distribution every value in a range [a, b] is equally likely. A random number generator's random() is uniform on [0, 1). It is rare in real data but common in simulation and in random search for hyperparameters.

Normal

The normal distribution is the familiar bell curve defined by a mean μ and standard deviation σ. The statistics for data science tutorial covers its shape and the 68-95-99.7 rule, so here we focus on using its CDF.

Python 3.8+ ships statistics.NormalDist, which gives you the PDF, CDF and inverse CDF with no extra install. Suppose exam scores are normal with mean 62 and SD 12.

from statistics import NormalDist

scores = NormalDist(mu=62, sigma=12)

print(f"P(score <= 50)        = {scores.cdf(50):.3f}")
print(f"P(score > 85)         = {1 - scores.cdf(85):.3f}")
print(f"P(70 <= score <= 80)  = {scores.cdf(80) - scores.cdf(70):.3f}")
print(f"Cut-off for top 10%   = {scores.inv_cdf(0.90):.1f}")

inv_cdf answers the reverse question: which score has 90% of students below it? This is how you set a cut-off for a scholarship or flag the top decile of customers.

Exponential: waiting time between events

If calls arrive as a Poisson process with λ = 4 per hour, the waiting time between calls follows an exponential distribution with mean 1/λ = 15 minutes. Its CDF is P(T ≤ t) = 1 − e^(−λt). It is right-skewed: most gaps are short, a few are long.

import numpy as np

lam_per_min = 4 / 60
rng = np.random.default_rng(1)
gaps = rng.exponential(scale=1 / lam_per_min, size=50_000)

print(f"Simulated mean gap: {gaps.mean():.1f} min (theory: {1/lam_per_min:.0f})")
print(f"Simulated P(gap > 30 min): {(gaps > 30).mean():.3f}")
print(f"Formula   P(gap > 30 min): {np.exp(-lam_per_min * 30):.3f}")

The simulation and the formula agree, which is a quick way to check any formula you are unsure of.

How to choose the right distribution

Start with "counted or measured?", then ask what exactly is being counted or measured.

Follow these steps for a new column:

1. Decide whether it is counted (discrete) or measured (continuous).
2. For counts, ask whether there is a fixed number of trials. Yes points to binomial; no points to Poisson.
3. For measurements, plot a histogram. Symmetric and bell-shaped suggests normal; waiting times suggest exponential; long right tails, like income or rent, often fit a log-normal.
4. Compare the data's mean and variance with what the candidate distribution predicts.
5. Simulate from the fitted distribution and check that the fake data looks like the real data.

To draw the histogram in step 3, you would use matplotlib on your laptop:

import numpy as np
import matplotlib.pyplot as plt

rng = np.random.default_rng(0)
calls = rng.poisson(lam=4, size=1000)

values, counts = np.unique(calls, return_counts=True)
plt.bar(values, counts / counts.sum(), color="#2563eb")
plt.xlabel("Calls per hour")
plt.ylabel("Probability")
plt.title("Simulated Poisson(4) call counts")
plt.show()

The runnable version below does the same checks with numbers instead of a picture, which is also how you would test a column inside a script.

import numpy as np

rng = np.random.default_rng(0)
calls = rng.poisson(lam=4, size=1000)          # pretend this is real data
values, counts = np.unique(calls, return_counts=True)

print("mean =", calls.mean().round(2), " variance =", calls.var(ddof=1).round(2))
for v, c in zip(values, counts):
    print(f"{v:>2} calls: {c / calls.size:.3f}  " + "#" * (c // 10))

Mean and variance both come out close to 4, which is exactly the Poisson signature from the tip above. The next topic, the central limit theorem, explains why the normal distribution keeps appearing even when the raw data is binomial or Poisson.

Common mistakes with distributions

• Assuming everything is normal. Counts, waiting times and money are usually skewed. Check the histogram first.
• Asking for P(X = x) on continuous data. Use a range or the CDF instead.
• Using binomial when trials are not independent. If one caller's decision affects the next, the binomial formula is off.
• Ignoring overdispersion. Count data with variance much bigger than the mean is not Poisson.
• Confusing PDF height with probability. A PDF value can exceed 1; only the area under it is a probability.

Key takeaways

• A distribution gives the probability of every possible value; probabilities add up to 1.
• Discrete variables use a PMF, continuous ones use a PDF, and both have a CDF.
• Binomial counts successes in n fixed trials; Poisson counts events in a window; exponential measures the wait between Poisson events.
• math.comb, statistics.NormalDist and numpy's random generator cover these without scipy.
• Choose a distribution by the data's story, then confirm with mean, variance and a simulation.

If you would like a mentor to review your distribution-fitting work on real projects, Debugshala's mentor-led Data Science & AI batches include that kind of guided practice. For quick revision, try probability MCQs on Debugshala Test.