⭐ Independent Events
Two events are independent when one happening does not change the probability of the other.
That’s the whole heart of it.
📌 Formal definition
Events 
This equation is the mathematical way of saying:
“The chance that both happen is just the product of their individual chances — because neither affects the other.”
🎯 Intuition
Think of two events that live in completely separate worlds.
- Flipping a coin
- Rolling a die
The coin doesn’t care what the die does.
The die doesn’t care what the coin does.
They’re blissfully unaware of each other.
That’s independence.
🧠 Another way to express it
Events
and
Meaning: even after you learn that one event happened, the probability of the other stays exactly the same.
🎲 Classic examples
- Drawing a card from a deck with replacement
- Weather tomorrow vs. rolling a die today
- Whether a student is left‑handed vs. whether they own a dog
None of these influence each other.
🚫 What independence is not
Students often confuse independence with mutually exclusive events.
But mutually exclusive events cannot be independent unless one has probability 0.
Why?
If two events can’t happen together, then knowing one happened tells you the other is impossible — which is the opposite of independence.