Independent vs. Mutually Exclusive

🎯 Mutually Exclusive (Disjoint) Events

Two events are mutually exclusive if they cannot happen at the same time.

• If one happens, the other is automatically impossible.
• Mathematically:
  

Example:
Rolling a die:

• “roll a 2”
• “roll a 5”
  You can’t roll a 2 and a 5 on the same single roll, so they’re mutually exclusive.

🎯 Independent Events

Two events are independent if knowing one happened gives you no information about the other.

• One event does not affect the probability of the other.
• Mathematically:
  

Example:

• “flip a coin and get heads”
• “roll a die and get a 6”
  The coin doesn’t care about the die.

🚨 The Key Insight

Mutually exclusive events cannot be independent (unless one of them has probability 0).

Why?

• If events are mutually exclusive, then .
• But if they were independent, we’d need .
• The only way both can be true is if , meaning at least one event has probability 0.

So for any real events with positive probability:

👉 Mutually exclusive = dependent
because if A happens, B becomes impossible.

🧠 Quick Comparison Table

Feature	Mutually Exclusive	Independent
Can they happen together?	No	Yes
Does one affect the other?	Yes (completely)	No
Formula
Typical example	One roll of a die	Coin flip + die roll

A playful way to remember

Mutually exclusive events are like two people who refuse to be in the same room.
Independent events are like two people who don’t even know the other exists.