Other Discrete Distributions

🎲 The Secret World of Special Probability Boxes

Imagine you have different magical boxes, each with its own special rules for picking things out. Let’s discover four amazing boxes that mathematicians use every day!

🌟 Our Adventure Map

We’re going to explore four special “probability boxes”:

1. Hypergeometric Box - Picking without putting back
2. Multinomial Box - Many colors, many picks
3. Discrete Uniform Box - Everyone gets an equal chance
4. Discrete Uniform Stats - The math behind equal chances

🐟 The Hypergeometric Distribution

The Story of the Fish Pond

Imagine a small pond with 10 fish. Some fish are gold (let’s say 4), and the rest are silver (6 fish).

You want to catch 3 fish with a net. Once you catch a fish, it’s in your bucket - you can’t throw it back!

Key Idea: When you pick items WITHOUT replacing them, and you care about how many “special” items you get, that’s the Hypergeometric distribution!

Why Is This Different?

Think about it:

• First grab: 4 gold fish out of 10 total
• If you caught a gold fish, now there are only 3 gold fish left out of 9!
• Each grab changes what’s left in the pond

The Magic Formula

P(X = k) = C(K,k) × C(N-K, n-k) / C(N,n)

What do these letters mean?

• N = Total items (10 fish)
• K = Special items (4 gold fish)
• n = How many you pick (3 fish)
• k = How many special ones you want

🎯 Real-Life Example

Lottery Ticket Check: A box has 20 tickets: 5 are winners, 15 are losers. You pick 4 tickets (no putting back).

What’s the chance you get exactly 2 winners?

N = 20, K = 5, n = 4, k = 2

P(X = 2) = C(5,2) × C(15,2) / C(20,4)
         = 10 × 105 / 4845
         = 1050 / 4845
         ≈ 0.217 or about 22%

📊 When to Use It

graph TD
    A["Are you picking items?"] --> B{Without replacement?}
    B -->|Yes| C{Counting special items?}
    C -->|Yes| D["Use Hypergeometric!"]
    B -->|No| E["Try Binomial instead"]

🌈 The Multinomial Distribution

The Story of the Candy Jar

Imagine a HUGE jar of candies with three colors:

• 🔴 Red (50%)
• 🟢 Green (30%)
• 🔵 Blue (20%)

You reach in and grab 10 candies (with replacement - magic jar refills!).

Key Idea: When you have MORE than 2 outcomes and you’re counting how many of EACH type you get, that’s Multinomial!

From Binomial to Multinomial

You already know the Binomial: heads or tails, win or lose - just 2 choices.

Multinomial is its bigger sibling: 3 or more choices!

The Magic Formula

P(X₁=n₁, X₂=n₂, ..., Xₖ=nₖ) =
    n! / (n₁! × n₂! × ... × nₖ!)
    × p₁^n₁ × p₂^n₂ × ... × pₖ^nₖ

Breaking it down:

• n = Total picks
• nᵢ = How many of type i
• pᵢ = Probability of type i
• All the n’s must add up to n!

🎯 Real-Life Example

Rolling a Die 6 Times: What’s the chance of getting exactly 2 ones, 2 twos, and 2 threes?

Each face has probability 1/6.

n = 6, n₁ = 2, n₂ = 2, n₃ = 2
p₁ = p₂ = p₃ = 1/6

P = 6!/(2!×2!×2!) × (1/6)² × (1/6)² × (1/6)²
  = 720/(2×2×2) × (1/6)⁶
  = 90 × (1/46656)
  = 90/46656
  ≈ 0.00193 or about 0.2%

📊 The Family Tree

graph TD
    A["Bernoulli<br>1 trial, 2 outcomes"] --> B["Binomial<br>n trials, 2 outcomes"]
    B --> C["Multinomial<br>n trials, k outcomes"]
    style C fill:#FFD700

🎰 The Discrete Uniform Distribution

The Story of the Fair Spinner

Imagine a spinner divided into 6 equal sections numbered 1 to 6 - just like a die!

Every number has the EXACT same chance of being picked.

Key Idea: When every outcome is equally likely, that’s Discrete Uniform!

The Simplest Formula Ever

If you have n equally likely outcomes:

P(X = x) = 1/n

That’s it! So simple and beautiful! 🎉

🎯 Real-Life Examples

Example 1: Rolling a Fair Die

Outcomes: 1, 2, 3, 4, 5, 6
n = 6
P(any number) = 1/6 ≈ 16.67%

Example 2: Random Card Draw Drawing one card from a shuffled deck of 52:

P(any specific card) = 1/52 ≈ 1.92%

Example 3: Random Number Generator Computer picks a number from 1 to 100:

P(any number) = 1/100 = 1%

📊 What It Looks Like

Imagine a bar chart where EVERY bar is the same height:

Probability
    |
1/6 |  ▓▓  ▓▓  ▓▓  ▓▓  ▓▓  ▓▓
    |  ▓▓  ▓▓  ▓▓  ▓▓  ▓▓  ▓▓
    |  ▓▓  ▓▓  ▓▓  ▓▓  ▓▓  ▓▓
    +--1---2---3---4---5---6--→ Value

📊 Discrete Uniform Stats

Finding the Middle (Mean/Expected Value)

For numbers from a to b:

Mean = (a + b) / 2

Example: Die roll (1 to 6)

Mean = (1 + 6) / 2 = 3.5

Wait, 3.5? But a die can’t land on 3.5!

True! The mean is the “balance point” - if you rolled forever and averaged, you’d get 3.5.

How Spread Out? (Variance)

Variance = ((b - a + 1)² - 1) / 12

Example: Die roll

n = 6 (values 1 to 6)
Variance = (6² - 1) / 12
         = (36 - 1) / 12
         = 35 / 12
         ≈ 2.917

Standard Deviation

Just take the square root of variance!

Standard Deviation = √Variance
                   = √2.917
                   ≈ 1.71

🎯 Complete Example

Random month picker (1 to 12):

a = 1, b = 12, n = 12

Mean = (1 + 12) / 2 = 6.5

Variance = (12² - 1) / 12 = 143/12 ≈ 11.92

Std Dev = √11.92 ≈ 3.45

This tells us: on average you’d pick month 6.5 (between June and July), with typical spread of about 3-4 months.

📊 Summary Flow

graph TD
    A["Discrete Uniform"] --> B["All outcomes equal"]
    B --> C["P = 1/n"]
    A --> D["Calculate Stats"]
    D --> E["Mean = a+b / 2"]
    D --> F["Var = n²-1 / 12"]
    F --> G["SD = √Variance"]

🎯 Quick Comparison Table

🌟 You Did It!

You’ve now discovered four special probability distributions:

1. ✅ Hypergeometric - For picking without putting back
2. ✅ Multinomial - For counting multiple categories
3. ✅ Discrete Uniform - When everyone’s equally likely
4. ✅ Discrete Uniform Stats - Mean, variance, and spread

Remember: Each “box” has its own rules. Pick the right box for your problem, and probability becomes your superpower! 🦸‍♂️

“Probability is not about predicting the future - it’s about understanding the possibilities.”