🎲 Discrete Distributions: The Bernoulli Family
A Story of Yes-or-No Adventures
🌟 The Big Picture
Imagine you’re flipping a magical coin. Every flip is a tiny adventure with only two endings: success or failure. That’s the heart of Bernoulli distributions!
Think of it like this: Every time you knock on a door asking “Want to buy cookies?”, you get either YES 🎉 or NO 😢. The Bernoulli family of distributions helps us understand patterns in these yes-or-no adventures.
🪙 Bernoulli Distribution
What is it?
The simplest member of the family. One try. One chance. Success or failure.
Think of it like: Asking ONE friend if they want to play. They say yes or no. That’s it!
The Magic Formula
- P(success) = p (your chance of hearing “yes”)
- P(failure) = 1 - p (your chance of hearing “no”)
Example: Shooting a Basketball
You take ONE shot at the basket.
- p = 0.7 (you make 70% of your shots)
- P(making it) = 0.7 ✅
- P(missing) = 0.3 ❌
Key Facts
| Property | Value |
|---|---|
| Mean (μ) | p |
| Variance (σ²) | p(1-p) |
Why does this matter?
- Mean = Your expected “score” (1 for success, 0 for failure)
- If p = 0.7, you expect 0.7 on average from each try
🎯 Binomial Distribution
What is it?
What happens when you repeat a Bernoulli experiment many times?
Think of it like: Asking 10 friends if they want to play. How many say yes?
The Setup
- n = number of tries (asking 10 friends)
- p = probability of success each time (maybe 60% say yes)
- k = number of successes you want to count
The Formula
P(X = k) = C(n,k) × p^k × (1-p)^(n-k)
Where C(n,k) = n! / (k! × (n-k)!)
Example: Cookie Sales
You knock on 5 doors. Each person has a 40% chance of buying.
What’s the probability that exactly 2 people buy?
- n = 5 doors
- p = 0.4 (40% buy)
- k = 2 buyers
C(5,2) = 10
P(X=2) = 10 × 0.4² × 0.6³
P(X=2) = 10 × 0.16 × 0.216
P(X=2) = 0.3456 ≈ 34.6%
📊 Binomial Mean and Variance
Here’s where it gets beautiful:
| Property | Formula | Meaning |
|---|---|---|
| Mean (μ) | n × p | Expected successes |
| Variance (σ²) | n × p × (1-p) | Spread of results |
Example: Flipping a coin 100 times (p = 0.5)
- Mean = 100 × 0.5 = 50 heads expected
- Variance = 100 × 0.5 × 0.5 = 25
- Standard deviation = √25 = 5
So you’d typically get 50 ± 5 heads (roughly 45-55 most times).
⏳ Geometric Distribution
What is it?
How long until your FIRST success?
Think of it like: Keep knocking on doors until someone FINALLY says yes. How many doors did you knock on?
The Key Question
“How many tries until the first success?”
The Formula
P(X = k) = (1-p)^(k-1) × p
This means: Fail (k-1) times, then succeed once.
Example: Finding a Parking Spot
Each spot has a 20% chance of being empty (p = 0.2).
What’s the probability you find a spot on the 3rd try?
P(X = 3) = 0.8² × 0.2
P(X = 3) = 0.64 × 0.2
P(X = 3) = 0.128 = 12.8%
First two spots taken (0.8 × 0.8), third one free (0.2)!
📊 Geometric Mean and Variance
| Property | Formula | Meaning |
|---|---|---|
| Mean (μ) | 1/p | Expected tries until success |
| Variance (σ²) | (1-p)/p² | Spread of waiting time |
Example: If p = 0.25 (25% success rate)
- Mean = 1/0.25 = 4 tries expected
- Variance = 0.75/0.0625 = 12
- You expect to wait about 4 tries, but it varies a lot!
🎮 Real Life Geometric
- Rolling a die until you get a 6: mean = 6 rolls
- Calling customers until one answers: depends on answer rate
- Trying keys until one works: mean = (# of keys)/1
🎪 Negative Binomial Distribution
What is it?
How many tries until you get r successes?
Think of it like: You need 3 friends to say “yes” to your party. How many friends do you ask?
The Evolution
| Distribution | Question |
|---|---|
| Bernoulli | One try: success or fail? |
| Binomial | Fixed tries: how many successes? |
| Geometric | How long until FIRST success? |
| Negative Binomial | How long until r-th success? |
The Formula
P(X = k) = C(k-1, r-1) × p^r × (1-p)^(k-r)
Where k = total trials needed, r = successes wanted.
Example: Finding 3 Volunteers
Each person has 30% chance of volunteering (p = 0.3).
What’s the probability you find 3 volunteers in exactly 7 asks?
- You need r = 3 successes
- In k = 7 total tries
- That means 4 failures and 3 successes
- The 7th person MUST be a success
C(6,2) = 15
P(X=7) = 15 × 0.3³ × 0.7⁴
P(X=7) = 15 × 0.027 × 0.2401
P(X=7) ≈ 0.097 = 9.7%
📊 Negative Binomial Mean and Variance
| Property | Formula | Meaning |
|---|---|---|
| Mean (μ) | r/p | Expected tries for r successes |
| Variance (σ²) | r(1-p)/p² | Spread of total trials |
Example: Getting 5 heads (r=5) with fair coin (p=0.5)
- Mean = 5/0.5 = 10 flips expected
- Variance = 5 × 0.5/0.25 = 10
🗺️ The Family Tree
graph TD A["🪙 Bernoulli<br>One Trial"] --> B["🎯 Binomial<br>Fixed n trials<br>Count successes"] A --> C["⏳ Geometric<br>Until 1st success<br>Count trials"] C --> D["🎪 Negative Binomial<br>Until r successes<br>Count trials"] style A fill:#FFE66D style B fill:#4ECDC4 style C fill:#FF6B6B style D fill:#95E1D3
🧠 Memory Tricks
When to Use Each:
| Situation | Distribution |
|---|---|
| “One shot, did I make it?” | Bernoulli |
| “10 shots, how many made?” | Binomial |
| “Keep shooting until I score” | Geometric |
| “Keep shooting until I score 5” | Negative Binomial |
The Mean Patterns:
| Distribution | Mean | Remember It As… |
|---|---|---|
| Bernoulli | p | “My success rate” |
| Binomial | np | “Tries × success rate” |
| Geometric | 1/p | “Flip the probability!” |
| Neg. Binomial | r/p | “Goals ÷ success rate” |
🎉 You Did It!
You now understand the Bernoulli family of distributions:
- ✅ Bernoulli - The single yes/no question
- ✅ Binomial - Counting successes in fixed trials
- ✅ Geometric - Waiting for first success
- ✅ Negative Binomial - Waiting for multiple successes
These distributions are everywhere:
- Quality control (defective items)
- Medical trials (patients cured)
- Sports (shots made)
- Gaming (loot drops)
- Sales (deals closed)
The secret? It’s all about counting successes and failures in yes-or-no situations. Master these, and you’ve got powerful tools for understanding randomness! 🚀