Multiplication Theorem and Independent Events

Rearranging the definition gives the multiplication theorem:

\[P(A\cap B)=P(A)\,P(B\mid A),\qquad P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B).\]

Use it for successive draws without replacement: the second probability is computed from what remains.

Events \(A\) and \(B\) are independent if knowing one does not change the probability of the other: \(P(A\mid B)=P(A)\), equivalently \(P(A\cap B)=P(A)P(B)\). If \(A,B\) are independent, so are \(A\) and \(B'\), \(A'\) and \(B\), \(A'\) and \(B'\).

At least one: for independent events, \(P(\text{at least one})=1-P(A')P(B')\cdots\).

Do not confuse independent with mutually exclusive: if \(P(A),P(B)>0\) and \(A\cap B=\varnothing\), then \(P(A\cap B)=0\ne P(A)P(B)\), so they are not independent.