Question

If P (A ∪ B) = 0.8 and P (A ∩ B) = 0.3, then P $\left(\overline{A}\right)$ + P $\left(\overline{B}\right)$ =
(a) 0.3
(b) 0.5
(c) 0.7
(d) 0.9

Answer 1

$$\begin{gathered} \left(\mathrm{d}\right)0.9 \\ P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \\ \Rightarrow P\left(A\right)+P\left(B\right)=P\left(A\cup B\right)+P\left(A\cap B\right) \\ \Rightarrow P\left(A\right)+P\left(B\right)=0.8+0.3 \\ \Rightarrow P\left(A\right)+P\left(B\right)=1.1 \\ \Rightarrow 1-P\left(\overline{A}\right)+1-P\left(\overline{B}\right)=1.1 \\ \Rightarrow P\left(\overline{A}\right)+P\left(\overline{B}\right)=2-1.1 \\ \Rightarrow P\left(\overline{A}\right)+P\left(\overline{B}\right)=0.9 \end{gathered}$$