Binomial & Poisson Distributions
Master event-based probability modeling for discrete events - from Bernoulli trials to rare event analysis.
Discrete Probability Models
Intermediate Level
40 min
π― Discrete Probability Distributions
Why These Distributions?
Binomial and Poisson distributions model count-based events in real-world scenarios.
They answer: "How many successes in n trials?" or "How many events in a time period?"
Real World Applications
- Click-through rate prediction
- Manufacturing defect analysis
- Customer arrival modeling
- Network failure prediction
π Distribution Comparison
B(n,p)
Binomial
Fixed trials, success/failure
vs
P(Ξ»)
Poisson
Rare events in time/space
Data Science Insight:
These distributions form the foundation for A/B testing, quality control, and reliability engineering in data science.
These distributions form the foundation for A/B testing, quality control, and reliability engineering in data science.
π² Bernoulli Trial
Definition
A Bernoulli trial is a random experiment with exactly two possible outcomes:
- Success (with probability p)
- Failure (with probability q = 1-p)
Properties
- Only two outcomes
- Probability p constant
- Trials are independent
- Example: Coin toss
Mathematical Model
X = {1 with probability p
0 with probability 1-p}
0 with probability 1-p}
π― Real World Bernoulli Examples
π―
Click Prediction
User clicks ad (p=0.02) or doesn't (0.98)
π₯
Medical Test
Test positive (p=0.05) or negative (0.95)
π
Quality Check
Item defective (p=0.01) or good (0.99)
π°
Loan Default
Customer defaults (p=0.03) or pays (0.97)
π Bernoulli Distribution (p = 0.3)
q=0.7
Failure (0)
p=0.3
Success (1)
Expected Value: E[X] = p = 0.3 | Variance: Var(X) = p(1-p) = 0.21
π’ Binomial Distribution
π What is Binomial Distribution?
B(n,p)
n trials, p success probability
Definition: Probability distribution of number of successes in n independent Bernoulli trials.
π Binomial Probability Formula
P(X = k) = nCk Γ pk Γ (1-p)n-k
n
Number of trials
k
Number of successes
p
Success probability
nCk
Combinations
π― Example: Coin Toss
Scenario: Toss fair coin 5 times. What's probability of exactly 3 heads?
Parameters:
- n = 5 (trials)
- k = 3 (successes = heads)
- p = 0.5 (probability of heads)
nCk
= 5C3 = 10
Γ
pα΅
= 0.5Β³ = 0.125
Γ
(1-p)βΏβ»α΅
= 0.5Β² = 0.25
=
10 Γ 0.125 Γ 0.25 = 0.3125
P(3 heads in 5 tosses) = 31.25%
β When to Use Binomial
- Fixed number of trials (n)
- Each trial has two outcomes
- Constant probability (p)
- Independent trials
- Counting successes
β Not Binomial
- Variable number of trials
- More than two outcomes
- Changing probability
- Dependent trials
- Continuous outcomes
π Binomial Distribution Properties
ΞΌ
Mean
np
np
ΟΒ²
Variance
np(1-p)
np(1-p)
Ο
Std Dev
β[np(1-p)]
β[np(1-p)]
π
Shape
Symmetric if p=0.5
Symmetric if p=0.5
π Poisson Distribution
π Modeling Rare Events
P(Ξ»)
Ξ» = average rate
Definition: Probability distribution of number of events occurring in a fixed interval of time/space, when events occur with known constant rate and independently.
π Poisson Probability Formula
P(X = k) = (Ξ»k Γ e-Ξ») / k!
Ξ» (lambda)
Average event rate
k
Number of events
e
Euler's number β 2.718
k!
k factorial
π― Example: Call Center
Scenario: Call center receives average 3 calls per hour. What's probability of exactly 5 calls in next hour?
Parameters:
- Ξ» = 3 (average calls per hour)
- k = 5 (events we want)
- e = 2.71828 (Euler's number)
Ξ»α΅
= 3β΅ = 243
Γ
eβ»Ξ»
= eβ»Β³ β 0.0498
/
k!
= 5! = 120
=
(243 Γ 0.0498) / 120 β 0.1008
P(5 calls in hour) β 10.08%
β When to Use Poisson
- Events occur independently
- Average rate (Ξ») is constant
- Events are rare
- Time/space based counting
- No fixed number of trials
π Real Examples
- Website visits per minute
- Emails received per hour
- Manufacturing defects per day
- Accidents per month
- Customer arrivals per hour
π Poisson Distribution Properties
ΞΌ
Mean
Ξ»
Ξ»
ΟΒ²
Variance
Ξ»
Ξ»
Ο
Std Dev
βΞ»
βΞ»
π
Unique Property
Mean = Variance
Mean = Variance
βοΈ Binomial vs Poisson: When to Use Which?
π Side-by-Side Comparison
| Aspect | Binomial | Poisson |
|---|---|---|
| Type | Counting successes in n trials | Counting events in time/space |
| Parameters | n (trials), p (success prob) | Ξ» (average rate) |
| Trials | Fixed number (n) | Not fixed |
| Outcomes | Success/Failure | Event count |
| When to Use | Yes/No questions, fixed trials | Rare events, time-based |
π― Decision Flowchart
Fixed number of trials?
β Yes
Only two outcomes?
β Yes
Use BINOMIAL
Counting events in time/space?
β Yes
Events independent & rare?
β Yes
Use POISSON
π Binomial β Poisson Approximation
Rule: When n is large (n β₯ 20) and p is small (p β€ 0.05), Binomial(n,p) β Poisson(Ξ» = np)
Binomial Example
n = 100, p = 0.02
P(X=3) using Binomial
P(X=3) using Binomial
Poisson Approximation
Ξ» = np = 100Γ0.02 = 2
P(X=3) using Poisson(2)
P(X=3) using Poisson(2)
π‘ Why it works: With large n and small p, events become rare β Poisson applies
π§ Data Science Applications
π―
Click Prediction
Problem: Predict clicks on website banner
Binomial Approach:
- n = 1000 impressions
- p = 0.02 (historical CTR)
- P(k clicks) = Binomial(1000, 0.02)
Business Use: Optimize ad placement based on expected clicks
β οΈ
Failure Analysis
Problem: Server failures in data center
Poisson Approach:
- Ξ» = 0.5 failures per day
- Time period = 7 days
- P(β₯3 failures) = 1 - P(0,1,2)
Business Use: Plan maintenance schedules, spare parts inventory
π
Quality Control
Problem: Defective items in manufacturing
Binomial
Batch sampling
Poisson
Continuous defects
Example: If 1% defect rate, what's P(β€2 defects in 100 items)?
π’ Case Study: E-commerce Website
π― Problem
Predict conversion rate and server load
π Binomial Application
Conversion rate modeling:
P(purchase|visit) = 0.03
Expected purchases in 1000 visits
π Poisson Application
Visitor arrivals:
Ξ» = 50 visitors/hour
Probability of >60 visitors
Real Implementation:
β’ Use Binomial for A/B testing (conversion rates)
β’ Use Poisson for infrastructure planning (peak loads)
β’ Use Binomial for A/B testing (conversion rates)
β’ Use Poisson for infrastructure planning (peak loads)
π― Interview Questions Preview
Q: When would you use Poisson instead of Binomial?
A: When events are rare, occur in time/space, and you don't have fixed number of trials (e.g., customer arrivals, defects)
Q: Hospital gets average 2 emergencies/hour. What's P(exactly 3 in next hour)?
A: Poisson with Ξ»=2, k=3: P(X=3) = (2Β³ Γ eβ»Β²)/3! = (8 Γ 0.1353)/6 β 0.1804 (18.04%)
β Chapter Summary & Cheatsheet
π²
Bernoulli Trial
Single trial, two outcomes, success probability p
E[X] = p, Var(X) = p(1-p)
π’
Binomial
n independent Bernoulli trials
P(X=k) = nCk pα΅(1-p)βΏβ»α΅
π
Poisson
Rare events in time/space
P(X=k) = (Ξ»α΅ eβ»Ξ»)/k!
β‘ Quick Decision Guide
Fixed trials β Binomial
Rare events β Poisson
nβ₯20, pβ€0.05 β Poisson approx
Click rate β Binomial
Failure rate β Poisson
π Must-Know Formulas
Binomial
P(X=k) = nCk pα΅(1-p)βΏβ»α΅
Poisson
P(X=k) = (Ξ»α΅ eβ»Ξ»)/k!
Mean
Bin: np | Pois: Ξ»
Variance
Bin: np(1-p) | Pois: Ξ»