Question

X and Y are two random independent events. It is known that P(X) = 0.40 and $P(X\cup Y^C)=\mathrm{0.7.}$ Which one of the following is the value of $P(X\cup Y)?$

• A. 0.7
• B. 0.5
• C. 0.4
• D. 0.3

Answer 1

The correct option is A 0.7

P(X) = 0.4 & $P(X\cup Y^c)=0.7$

$P(X\cup Y^c)=P\left(X\right)+P\left(Y^c\right)-P(X\cap Y^c)$

$0.7=P\left(X\right)+P\left(Y^c\right)-P\left(X\right).P\left(Y^c\right)$
$0.7=0.4+P\left(Y^c\right)-0.4P\left(Y^c\right)$

$\Rightarrow 0.6P\left(Y^c\right)=0.3$
$\Rightarrow P\left(Y^c\right)=0.5$
$\Rightarrow 1-P\left(Y\right)=\left(0.5\right)$

$\Rightarrow P\left(Y\right)=0.5$

$\Rightarrow P(X\cup Y)=P\left(X\right)+P\left(Y\right)-P(X\cap Y)$

$=0.4+0.5-0.4\times 0.5$

$\Rightarrow P(X\cup Y)=0.7$