PROBABILITY 04
Conditional Probability
Conditional probability describes probability after information is known to have occurred.
The main definition
If \(P(B)>0\),
\[\boxed{P(A\mid B)=\frac{P(A\cap B)}{P(B)}}\]
Read \(P(A\mid B)\) as “the probability of \(A\) given \(B\).”
1. Why the denominator changes
Once \(B\) is known, the possible outcomes are restricted to \(B\). Conditional probability measures how much of that restricted probability belongs to \(A\).
2. Multiplication rule
Rearranging the definition gives
\[\boxed{P(A\cap B)=P(A\mid B)P(B)}.\]Equally,
\[P(A\cap B)=P(B\mid A)P(A).\]3. Example
For a fair die, let \(B\) be “the result is even” and \(A\) be “the result is greater than 3.” Then
\[B=\{2,4,6\},\qquad A\cap B=\{4,6\},\]so
\[P(A\mid B)=\frac{2/6}{3/6}=\frac23.\]4. Bayes' theorem
Combining the two multiplication forms gives
\[\boxed{P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}}\]
Bayes' theorem reverses the direction of conditioning.
5. Conditioning changes the probability space
\[\boxed{\text{new information}\rightarrow\text{restricted possibilities}\rightarrow\text{updated probability}}\]