Question

If A and B are two events, then which of the following does not represent the probability of at most one of A,B occurs.

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

Answer 1

Answer: (D)

$P(A\cap \overline{B})+P(\overline{A}\cap \overline{B})-P(\overline{A}\cap \overline{B})$

Probability of occurence of at most one event $=1-P(A\cap B)$

$=P(A\cap \overline{B})+P(\overline{A}\cap B)+P(\overline{A}\cap \overline{B})$

As we know that,

$P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)$

$P\left(A\right)=1-P\left(\overline{A}\right)$

Therefore,

$1-P(A\cap B)=1-[(1-P\left(\overline{A}\right))+(1-P\left(\overline{B}\right))-P(A\cup B)]$

$=P\left(\overline{A}\right)+P\left(\overline{B}\right)+P(A\cup B)-1$

$=P\left(\overline{A}\right)+P\left(\overline{B}\right)-P(\overline{A}\cap \overline{B})$

Therefore, among the given option $\left(D\right)$ does not represent the probability of at most one of $A,B$ occurs.

Hence $\left(D\right)$ is the correct answer.