Question

Let $A$ and $B$ any two independent events with $0<P\left(A\right)<1$ and $0<P\left(B\right)<1.$ Which of the following statements are TRUE?

$\text{I. P(A occurs but B does not occur})=P(A)-P(A\cap B)$
$\text{II. P(Exactly one of A and B occurs})=P(A)+P(B)-P(A\cap B)$
$\text{III. P(Neither A nor B occurs})=(P\left(A\right)-1\left)\right(P\left(B\right)-1)$
$\text{IV. P(A occurs given that B does not occur})=P(A)$

• A. Only $\text{I, II, III}$
• B. Only $\text{I, II, IV}$
• C. Only $\text{I, III, IV}$
• D. All of them

Answer 1

The correct option is C Only $\text{I, III, IV}$

We know that if $A$ and $B$ are independent, then
$1.$ $A$ and $B^C$ are independent.
$2.$ $A^C$ and $B$ are independent.
$3.$ $A^C$ and $B^C$ are independent.

$\text{I. P(A occurs but B does not occur})$
$=P(A\cap B^C)$
$=P\left(A\right)P\left(B^C\right)$
$=P\left(A\right)(1-P(B\left)\right)$
$=P\left(A\right)-P\left(A\right)P\left(B\right)$
$=P\left(A\right)-P(A\cap B)$

$\text{II. P(Exactly one of A and B occurs})$
$=P\left(\right(A\cap B^C)\cup (A^C\cap B\left)\right)$
$=P\left(A\right)P\left(B^C\right)+P\left(A^C\right)P\left(B\right)$
$=P\left(A\right)+P\left(B\right)-2P(A\cap B)$

$\text{III. P(Neither A nor B occurs})$
$=P(A^C\cap B^C)$
$=P\left(A^C\right)P\left(B^C\right)$
$=(1-P(A\left)\right)(1-P(B\left)\right)$
$=\left(P\right(A)-1)\left(P\right(B)-1)$

$\text{IV. P(A occurs given that B does not occur})$
$=P\left(A|B^C\right)$
$=\frac{P(A\cap B^C)}{P\left(B^C\right)}$
$=\frac{P\left(A\right)P\left(B^C\right)}{P\left(B^C\right)}=P\left(A\right)$
Alternatively, we can directly conclude that $P\left(A|B^C\right)=P\left(A\right)$ as $A$ and $B^C$ are independent.