PROBABILITY 05
Independence
Two events are independent when knowing that one occurred does not change the probability of the other.
The definition
This equation defines independence of events \(A\) and \(B\).
1. Conditional interpretation
If \(P(B)>0\), independence implies
\[P(A\mid B)=P(A).\]Learning that \(B\) occurred gives no probability information about \(A\).
2. Example
For two independent fair coin tosses, let \(A\) be “first toss is heads” and \(B\) be “second toss is heads.” Then
\[P(A\cap B)=\frac14=\frac12\frac12.\]3. Independence is not disjointness
Disjoint events cannot occur together. Independent events can occur together, but one does not affect the probability of the other.
If nonzero-probability events are disjoint, they cannot be independent because
\[P(A\cap B)=0\ne P(A)P(B).\]4. Independent random variables
Later, random variables \(X\) and \(Y\) are independent when their joint distribution factorises into their separate distributions.
5. Why independence matters
Independence allows complicated joint probabilities and expectations to simplify. But it must be justified by the model rather than assumed merely for convenience.