# Difference between revisions of "TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2019W/1.3"

Independence

Let A and B be independent events.

(a) Prove that $A^c$ and $B^c$ are also independent.
(b) If we additionally know that $P(A|B) = 0.6$ and $P(B|A) = 0.3$, compute the probabilities of the following events
(i) at most one of A or B
(ii) either A or B but not both.

## Lösungsvorschlag

Beweis für (a) von StackExchange:

\begin{align} P(A^c \cap B^c) &= 1 - P(A \cup B) \\ &= 1 - P(A) - P(B) + P(A \cap B) \\ &= 1 - P(A) - P(B) + P(A)P(B) \\ &= (1-P(A))(1-P(B)) \\ &= P(A^c)P(B^c). \end{align}

Siehe auch diese PDF von math.dartmouth.edu.

(b): TBD