Let a and b be two independent events. if p(a) = 0.5, what can you say about p(a | b)?

$P(A \cup B) = P(A)+P(B)-P(A\cap B)$.

Given $P(A)=P(B)=x$,

so $x+x-(x\times x)=0.5\Rightarrow 2x-x^2=\frac{1}{2}\Rightarrow 4x-2x^2-1=0$.

We have to solve, $2x^2-4x+1=0$. Here, $a=2,b=-4,c=1$.

Now,$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{4\pm\sqrt{16-4\times2\times(-1)}}{2\times2}=\frac{4\pm\sqrt{8}}{4}=1\pm\frac{1}{\sqrt{2}}$$

We have $x=1.707 \& 0.292$.

$\because, x $ is probability $x {\not>} 2$.

$\therefore \boxed{x= P(A)= 0.292}$

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