Event Algebra

🎲 Event Algebra: The Magic of Probability Sets

Imagine you have a magical toy box. Some toys are inside, some are outside. Event Algebra is like learning the rules of what’s IN and what’s OUT of your toy box!

🌟 The Big Picture

Think of probability events like groups of toys. Sometimes you want:

• All the toys you DON’T have (complement)
• Toys from EITHER box (union)
• Toys in BOTH boxes (intersection)

Let’s learn these magic rules with a simple story!

🎯 Our Universe: The Sample Space

Before we start, meet our sample space - it’s like ALL possible toys that exist.

Example: Rolling a die

• Sample Space S = {1, 2, 3, 4, 5, 6}
• That’s ALL possible outcomes!

graph TD
    S["🎲 Sample Space S"]
    S --> A["1, 2, 3, 4, 5, 6"]

❌ Complement of an Event (A’)

What is it?

The complement of an event is everything that is NOT in that event.

Think of it like this:

If your friend picked the RED candies, the complement is ALL the candies that are NOT red!

Simple Example

• Event A = Rolling an even number = {2, 4, 6}
• Complement A’ = Rolling NOT even = {1, 3, 5}

graph TD
    S["Sample Space S = {1,2,3,4,5,6}"]
    S --> A["Event A = {2,4,6}<br>Even numbers"]
    S --> AC["A' = {1,3,5}<br>NOT even = Odd!"]

🔑 Key Insight

Whatever is NOT in A, is in A’. Simple as that!

⚖️ The Complement Rule

The Magic Formula

$P(A’) = 1 - P(A)$

What does this mean?

If you know how likely something IS, you automatically know how likely it ISN’T!

Real Example

Scenario: A bag has 10 marbles. 3 are blue.

• P(Blue) = 3/10 = 0.3
• P(NOT Blue) = 1 - 0.3 = 0.7

Why This Works

All probabilities must add up to 1 (100%).

• Either something happens OR it doesn’t!
• P(A) + P(A’) = 1 (always!)

graph TD
    T["Total Probability = 1"]
    T --> PA["P#40;A#41;"]
    T --> PAC["P#40;A'#41; = 1 - P#40;A#41;"]

🤝 Union of Events (A ∪ B)

What is it?

Union means “OR” — outcomes in A OR B OR both!

Think of it like this:

You can have pizza OR ice cream OR both at the party!

Visual Picture

The union is everything inside EITHER circle (or both).

Simple Example

• Event A = {1, 2, 3} (rolling 1, 2, or 3)
• Event B = {2, 3, 4} (rolling 2, 3, or 4)
• A ∪ B = {1, 2, 3, 4} (everything in either!)

graph TD
    A["A = {1, 2, 3}"]
    B["B = {2, 3, 4}"]
    U["A ∪ B = {1, 2, 3, 4}"]
    A --> U
    B --> U

🎨 Notice!

We don’t count 2 and 3 twice — they appear once in the union.

🎯 Intersection of Events (A ∩ B)

What is it?

Intersection means “AND” — outcomes in A AND B together!

Think of it like this:

You want toys that are BOTH red AND round. Only toys matching BOTH conditions count!

Visual Picture

The intersection is ONLY what’s in BOTH circles.

Simple Example

• Event A = {1, 2, 3}
• Event B = {2, 3, 4}
• A ∩ B = {2, 3} (only what’s in BOTH!)

graph TD
    A["A = {1, 2, 3}"]
    B["B = {2, 3, 4}"]
    I["A ∩ B = {2, 3}"]
    A --> I
    B --> I

🔮 De Morgan’s Laws

The Two Magic Rules

Augustus De Morgan discovered two powerful rules that connect complements, unions, and intersections!

Law 1: Complement of Union

$(A \cup B)’ = A’ \cap B’$

In words: “NOT (A or B)” = “NOT A AND NOT B”

Example:

• NOT (raining OR cold) = NOT raining AND NOT cold
• If it’s neither raining nor cold = it’s dry AND warm!

Law 2: Complement of Intersection

$(A \cap B)’ = A’ \cup B’$

In words: “NOT (A and B)” = “NOT A OR NOT B”

Example:

• NOT (tall AND heavy) = NOT tall OR NOT heavy
• If someone isn’t both tall AND heavy = they’re short OR light (or both)!

graph TD
    L1["Law 1: #40;A∪B#41;' = A'∩B'"]
    L2["Law 2: #40;A∩B#41;' = A'∪B'"]
    L1 --> R1["NOT-OR becomes AND-NOT"]
    L2 --> R2["NOT-AND becomes OR-NOT"]

🧠 Memory Trick

When you “push” the complement inside:

• Union (∪) becomes Intersection (∩)
• Intersection (∩) becomes Union (∪)

🚫 Mutually Exclusive Events

What is it?

Two events are mutually exclusive if they CANNOT happen together.

Think of it like this:

You can’t be in the living room AND bedroom at the same time!

The Rule

$A \cap B = \emptyset$ (The intersection is EMPTY — nothing is in both!)

Simple Example

• Event A = Rolling an even number = {2, 4, 6}
• Event B = Rolling an odd number = {1, 3, 5}
• A ∩ B = {} (empty! You can’t roll even AND odd!)

graph TD
    A["Even: {2,4,6}"]
    B["Odd: {1,3,5}"]
    N["No overlap!"]
    A --> N
    B --> N

🎁 Special Property

For mutually exclusive events: $P(A \cup B) = P(A) + P(B)$

No need to subtract anything — there’s no overlap!

🌐 Exhaustive Events

What is it?

Events are exhaustive if they cover ALL possibilities together.

Think of it like this:

If you divide a pizza among friends, everyone together has the WHOLE pizza!

The Rule

$A_1 \cup A_2 \cup … \cup A_n = S$ (All events together = the entire sample space!)

Simple Example

• Event A = Rolling 1, 2, or 3 = {1, 2, 3}
• Event B = Rolling 4, 5, or 6 = {4, 5, 6}
• A ∪ B = {1, 2, 3, 4, 5, 6} = S (the whole sample space!)

These are exhaustive — they cover EVERY possibility!

graph TD
    S["Sample Space S"]
    A["A = {1,2,3}"]
    B["B = {4,5,6}"]
    A --> C["A ∪ B = S ✓"]
    B --> C

🔑 Key Insight

Exhaustive events guarantee that at least one of them MUST happen!

🎭 Putting It All Together

The Complete Picture

🚀 Real-World Applications

Weather Forecast

• Event R = It rains
• Event R’ = It doesn’t rain (complement)
• P® = 30% means P(R’) = 70%

Playing Cards

• Event H = Drawing a Heart
• Event K = Drawing a King
• H ∩ K = King of Hearts (AND)
• H ∪ K = Any Heart OR any King (OR)

Rolling Two Dice

• Mutually Exclusive: Getting sum 7 AND sum 11 (can’t happen together)
• Exhaustive: All possible sums (2 through 12) cover everything

💡 Key Takeaways

1. Complement (A’): Everything NOT in A. P(A’) = 1 - P(A)

2. Union (A ∪ B): Everything in A OR B (or both)

3. Intersection (A ∩ B): Only what’s in BOTH A AND B

4. De Morgan’s Laws:

   • (A ∪ B)’ = A’ ∩ B’
   • (A ∩ B)’ = A’ ∪ B’

5. Mutually Exclusive: No overlap (A ∩ B = ∅)

6. Exhaustive: Cover everything (A ∪ B ∪ … = S)

🎉 You Did It!

You now understand the fundamental building blocks of probability! These concepts are like LEGO pieces — once you master them, you can build anything in probability theory!

Remember:

• Complement = NOT
• Union = OR
• Intersection = AND
• Mutually Exclusive = Never together
• Exhaustive = Always covers all

Now you’re ready to play with probabilities like a pro! 🎲✨