Question

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

• A. $1-P(A\cap B)$
• B. $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)-P(A^{\prime }\cap B^{\prime })$
• C. $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)+P(A^{\prime }\cup B^{\prime })-1$
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $1-P(A\cap B)$
B $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)-P(A^{\prime }\cap B^{\prime })$

The shaded portion in the illustration is the required region.
(a) This obviously satisfies.
(b) $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)-P(A^{\prime }\cap B^{\prime })=P(A^{\prime }\cap B)+P(A^{\prime }\cap B^{\prime })+P(A\cap B^{\prime })=1-P(A\cap B)$
(c) Comparing (b) with (c), we can see that it is not true as $P(A^{\prime }\cup B^{\prime })\ne 1-P(A^{\prime }\cap B^{\prime })$
(d) This obviously satisfies.
Hence, (a), (b), (d) are correct.