Question

The probability of the simultaneous occurrence of two events $A$ and $B$ is $p$. If the probability that exactly one of $A,B$ occurs is $q$, then

• A. $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)=2+2q-p$
• B. $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)=2-2p-q$
• C. $P(A\cap B|A\cup B)=\frac{p}{p+q}$
• D. $P(A^{\prime }\cap B^{\prime })=1-p-q$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)=2-2p-q$
C $P(A\cap B|A\cup B)=\frac{p}{p+q}$
D $P(A^{\prime }\cap B^{\prime })=1-p-q$

$P\left(A^{\prime }\right)+P\left(B^{\prime }\right)=a+b+2r=q+2r=1-p+r=1-p+(1-p-q)=2-2p-q.$
$P(A\cap B|A\cup B)=p/(a+p+b)=p/(p+q)$
$P(A^{\prime }\cap B^{\prime })=r=1-(a+b+p)=1-q-p$
Hence, (b), (c), (d) are correct.