Addition Rules

🎲 The Magic of Adding Chances: Probability Addition Rules

The Cookie Jar Story 🍪

Imagine you have a magical cookie jar. Inside are 10 cookies: 4 chocolate chip, 3 oatmeal, 2 peanut butter, and 1 rainbow cookie.

You close your eyes and reach in. What are the chances you’ll grab something yummy? Spoiler: 100% — they’re all yummy!

But what if you want a chocolate chip OR an oatmeal cookie? That’s where Addition Rules come in!

🌟 The General Addition Rule

The Big Idea

When you want to find the chance of this OR that happening, you ADD the chances together. But there’s a sneaky trick — if both things can happen at the same time, you counted them twice!

The Formula

P(A or B) = P(A) + P(B) - P(A and B)

Think of it like counting friends at a party:

• 🎂 Friends who like cake = 6
• 🍦 Friends who like ice cream = 5
• 🎂🍦 Friends who like BOTH = 3

If you just add 6 + 5 = 11, you counted those 3 friends twice! So: 6 + 5 - 3 = 8 friends like cake OR ice cream.

Real Example 🎴

A deck has 52 cards. What’s the chance of drawing a King OR a Heart?

• Kings: 4/52
• Hearts: 13/52
• King of Hearts (both!): 1/52

P(King or Heart) = 4/52 + 13/52 - 1/52
                 = 16/52
                 = 4/13 ≈ 30.8%

🔒 Mutually Exclusive Addition

The Big Idea

Sometimes, two things can NEVER happen together. Like being in London AND New York at the exact same moment. Impossible!

When events can’t overlap, they’re called mutually exclusive. No double-counting needed!

The Simple Formula

P(A or B) = P(A) + P(B)

That’s it! Just add. No subtraction needed.

Real Example 🎲

Rolling a die. What’s the chance of getting a 2 OR a 5?

Can you roll a 2 AND a 5 at the same time? Nope! They’re mutually exclusive.

P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3 ≈ 33.3%

Quick Check: Are They Mutually Exclusive?

🔍 The Inclusion-Exclusion Principle

The Big Idea

This is the General Addition Rule’s big sibling. It works for 2, 3, or even 100 events!

The pattern: Add everything, then subtract overlaps, then add back triple overlaps, then subtract…

For Three Events

P(A or B or C) = P(A) + P(B) + P(C)
               - P(A and B) - P(A and C) - P(B and C)
               + P(A and B and C)

The Venn Diagram Way

graph TD
    A["🔵 Event A"] --> O["Overlap Zone"]
    B["🟢 Event B"] --> O
    C["🟡 Event C"] --> O
    O --> R["Add once, not twice!"]

Real Example 🏫

In a class of 30 students:

• 📚 15 like Math
• 🔬 12 like Science
• 🎨 10 like Art
• 📚🔬 6 like Math AND Science
• 📚🎨 4 like Math AND Art
• 🔬🎨 3 like Science AND Art
• 📚🔬🎨 2 like ALL THREE

How many like at least one subject?

= 15 + 12 + 10 - 6 - 4 - 3 + 2
= 37 - 13 + 2
= 26 students

🎯 “At Least One” Probability

The Big Idea

“What’s the chance of getting at least one heads in 3 coin flips?”

Counting all the “at least one” cases is hard. But counting “ZERO” is easy!

The Clever Trick

P(at least one) = 1 - P(none)

Everything minus nothing = something!

Real Example 🪙

Three coin flips. What’s the chance of at least one heads?

The Hard Way: Count HHH, HHT, HTH, HTT, THH, THT, TTH… so many!

The Easy Way:

• P(no heads at all) = P(TTT) = 1/2 × 1/2 × 1/2 = 1/8

P(at least one heads) = 1 - 1/8 = 7/8 = 87.5%

Another Example 🎲

Roll a die 4 times. What’s the chance of getting at least one 6?

• P(not a 6 on one roll) = 5/6
• P(no 6s in 4 rolls) = (5/6)⁴ = 625/1296

P(at least one 6) = 1 - 625/1296
                  = 671/1296 ≈ 51.8%

🗺️ The Complete Map

graph TD
    Q{What type of OR?}
    Q -->|Can happen together| G["General Addition Rule"]
    Q -->|Can NEVER happen together| M["Mutually Exclusive"]
    Q -->|Multiple events overlapping| I["Inclusion-Exclusion"]
    Q -->|At least one success| A["Complement Method"]

    G --> F1["P#40;A∪B#41; = P#40;A#41; + P#40;B#41; - P#40;A∩B#41;"]
    M --> F2["P#40;A∪B#41; = P#40;A#41; + P#40;B#41;"]
    I --> F3["Add singles - pairs + triples..."]
    A --> F4["1 - P#40;none#41;"]

💡 Pro Tips

Tip 1: The Word “OR” = Addition

Whenever you see “or” in a probability question, think addition rules!

Tip 2: Draw a Venn Diagram

Circles overlapping? Use General Addition. Circles separate? Mutually Exclusive.

Tip 3: “At Least One” = Flip It

Always use: 1 - P(none happens). It’s faster every time!

Tip 4: Check Your Answer

Probabilities must be between 0 and 1. If you get 1.5 or -0.3, something’s wrong!

🎮 Summary Table

🌈 You’ve Got This!

You now know the secret language of “OR” in probability:

1. General Rule — Add and subtract overlaps
2. Mutually Exclusive — Just add (no overlap!)
3. Inclusion-Exclusion — The pattern for many events
4. At Least One — Use the complement trick

Next time someone asks “What are the chances?”, you’ll know exactly how to figure it out! 🎲✨