IDEA FOUND // IDEA 144
Probabilistic Independence
“Two events can occur together while leaving each other's probabilities unchanged.”
01 / PLAINLY
What it means, plainly
Two events are independent when learning that one occurred does not change the probability of the other. Independence describes a process or model, not opposing outcomes.
02 / CONTEXT
A little more
A and B are independent if learning one does not change the probability of the other; equivalently, P(A∩B)=P(A)P(B). This is a property of the model or process, not a visual impression.
03 / WHY IT MATTERS
Why it matters
It tells you when probabilities can be combined with the product rule and when one observation should update the other.
04 / EXAMPLE
A familiar example
Across two fair coin tosses, the first result does not change the probability of heads on the second.
05 / LIMIT
What it does not mean
Independent does not mean mutually exclusive; positive-probability events that exclude each other are dependent.
06 / NOTICE
Notice it in your day
Compare ‘heads on the first toss’ with ‘heads on the second,’ then with ‘tails on the first toss.’ Decide which pair can occur together.
FINAL NOTE
The idea worth keeping
Check the product rule or conditional probability instead of trusting intuition.
QUESTIONS / 02
Questions people still have
What does it mean for A and B to be independent?
Learning one does not change the probability of the other.
Does independent mean mutually exclusive?
No. Independent events can occur together; positive-probability events that exclude each other are dependent.