Exercise 1.45

Question

Let $A$ and $B$ be events. The symmetric difference $A 	riangle B$ is defined to be the set of all elements that are in $A$ or $B$ but not both. In logic and engineering, this event is also called the $X O R$ (exclusive or) of $A$ and $B$. Show that
$$
P(A 	riangle B)=P(A)+P(B)-2 P(A \cap B)
$$
directly using the axioms of probability.

fifthist x · Accepted Answer

We have
$$B \bigtriangleup A = (A \cup B) - (A \cap B).$$
By \href{./exercise_1-44}{problem 44}, 
\begin{align*}
P(B \bigtriangleup A) & = P(A \cup B) - P(A \cap B) 
onumber && \
 & = P(A) + P(B) - P(A \cap B) - P(A \cap B) 
onumber && \
 & = P(A) + P(B) - 2P(A \cap B) 
onumber
\end{align*}

Exercise 1.45

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Exercise 1.45

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