Question

If $A$ and $B$ are two events such that $P\left(A\right)=\frac{3}{4}$ and $P\left(B\right)=\frac{5}{8}$, then

• A. $P(A\cup B)\ge \frac{3}{4}$
• B. $P(A^{\prime }\cap B)\le \frac{1}{4}$
• C. $\frac{3}{8}\le P(A\cap B)\le \frac{5}{8}$
• D. All of the above

Answer 1

The correct option is D All of the above

$A\subset A\cup B\Rightarrow P\left(A\right)\le P(A\cup B)$
$\Rightarrow P(A\cup B)\ge 3/4$
Also, $P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cup B)\ge P\left(A\right)+P\left(B\right)-1$
$=\frac{3}{4}+\frac{5}{8}-1=\frac{3}{8}$
Now, $A\cap B\subset B\Rightarrow P(A\cap B)\le P\left(B\right)=\frac{5}{8}$
$\therefore \frac{3}{8}\le P(A\cap B)\le \frac{5}{8}....\left(i\right)$
Next $P(A\cap B^{\prime })=P\left(A\right)-P(A\cap B)$
$\Rightarrow \frac{3}{4}-\frac{5}{8}\le P(A\cap B^{\prime })\le \frac{3}{4}-\frac{3}{8}$
$\Rightarrow \frac{1}{8}\le P(A\cap B^{\prime })\le \frac{3}{8}$
$\because P(A^{\prime }\cap B)=P\left(B\right)-P(A\cap B)$
$\therefore P(A\cap B)=P\left(B\right)-P(A^{\prime }\cap B^{\prime })$ [Using Eq. (i)]
$\Rightarrow \frac{3}{8}\le P\left(B\right)-P(A^{\prime }\cap B)\le \frac{5}{8}$
$\Rightarrow 0\le P(A^{\prime }\cap B)\le \frac{1}{4}$.