Probability - exactly one of A and B occurs

Question

$A$ and $B$ are events that are subsets of the sample space. $C$ is the event that exactly one of $A$ and $B$ occurs.

1) Write an expression for $C$ in terms of unions, intersections and complements involving the events $A$ and $B$

2) Let $P$ be a probability defined on the events of the sample space. Write an expression for $P(C)$ in terms of $P(A)$, $P(B)$ and $P(A \cap B)$. Give proof of your result.

Would I be right in saying that 1) is just

$C=(A \cup B)-(A \cap B)$

Or would it be? $(A \cap B^c) \cup (A^c \cap B)$

Answer 1

So you are looking for the shaded area:

1)

$C=(A \cap B^- )\cup (A^- \cap B)$

$B^-$ is the complement of B

or $C=(A\cup B)-(A∩B)$

2)

$P(C)=P(A∪B)−P(A∩B)$

Answer 2

Both are correct, but the first is not allowed because difference is not allowed in the problem statement.

Answer 3

I guess you are looking for $$P(C) = P(A \cup B) - P(A \cap B)$$

In the set $A\cup B$, either C happens or $A \cap B$ happens - these are mutually exclusive.

Answer 4

The probability that exactly one of $A$ and one of $B$ occurs is zero. All of the above solutions give the probability that only $A$ or only $B$ occurs.

Answer 5

Note that the answer to question 2 is just $$P(A\cup B)-P(A\cap B)=P(A\cup B)-(P(A)+P(B)-P(A\cup B))$$ $$P(A\cup B)-P(A\cap B)=2P(A\cup B)-P(A)-P(B)$$