Question

If A and B are two events. The probability that at most one of A, B occurs is

• A. $1-P(A\cap B)$
• B. $P\left(\overline{A}\right)+P\left(\overline{B}\right)-P(\overline{A}\cap B)$
• C. $P\left(\overline{A}\right)+P\left(\overline{B}\right)+P(\overline{A}\cap \overline{B})$
• D. All of the above

Answer 1

The correct option is A $1-P(A\cap B)$

C is the event that exactly one of the following events:

A but not B

B but not A

Not B and Not A (Neither A nor B)

1 is the event $A\setminus B$ or $A\cap B^C$

2 is the event $B\setminus A$or $B\cap A^C$

3 is the event $B^C\cap A^C$ (not B and not A) or $(A\cup B{)}^C(not\left(AorB\right))$

Observe that the events are mutually exclusive.

Hence,

$P\left(C\right)=P(A\cap B^C)+P(B\cap A^C)+P(B^C\cap A^C)$

$P\left(C\right)=P(A\cap B^C)+P(B\cap A^C)+P(B^C\cap A^C)$

$P\left(C\right)=1-P(A\cap B)$$

$P\left(C\right)=1-P\left(A\right)P\left(B\right)$