Question

Prove that if A and B are independent events then A conpliment and B compliment are also independent events

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}, \\ \mathrm{P}\left(\overline{\mathrm{A}}\cap \overline{\mathrm{B}}\right)=\mathrm{P}\left(\overline{\mathrm{A}\cup \mathrm{B}}\right) \\ \Rightarrow \mathrm{P}\left(\overline{\mathrm{A}}\cap \overline{\mathrm{B}}\right)=1-\mathrm{P}\left(\mathrm{A}\cup \mathrm{B}\right) \\ \Rightarrow \mathrm{P}\left(\overline{\mathrm{A}}\cap \overline{\mathrm{B}}\right)=1-\left[\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)-\mathrm{P}\left(\mathrm{A}\cap \mathrm{B}\right)\right] \\ \Rightarrow \mathrm{P}\left(\overline{\mathrm{A}}\cap \overline{\mathrm{B}}\right)=1-\left[\mathrm{P}\left(\mathrm{A}\right)+\mathrm{P}\left(\mathrm{B}\right)-\mathrm{P}\left(\mathrm{A}\right).\mathrm{P}\left(\mathrm{B}\right)\right] \\ \Rightarrow \mathrm{P}\left(\overline{\mathrm{A}}\cap \overline{\mathrm{B}}\right)=1-\mathrm{P}\left(\mathrm{A}\right)-\mathrm{P}\left(\mathrm{B}\right)+\mathrm{P}\left(\mathrm{A}\right).\mathrm{P}\left(\mathrm{B}\right) \\ \Rightarrow \mathrm{P}\left(\overline{\mathrm{A}}\cap \overline{\mathrm{B}}\right)=\left[1-\mathrm{P}\left(\mathrm{A}\right)\right]-.\mathrm{P}\left(\mathrm{B}\right)\left[1-\mathrm{P}\left(\mathrm{A}\right)\right] \\ \Rightarrow \mathrm{P}\left(\overline{\mathrm{A}}\cap \overline{\mathrm{B}}\right)=\left[1-\mathrm{P}\left(\mathrm{A}\right)\right]\left[1-\mathrm{P}\left(\mathrm{B}\right)\right]=\mathrm{P}\left(\overline{\mathrm{A}}\right).\mathrm{P}\left(\overline{\mathrm{B}}\right) \\ \mathrm{Hence},\overline{\mathrm{A}}\mathrm{and}\overline{\mathrm{B}}\mathrm{are}\mathrm{independent}\mathrm{events}. \end{gathered}$$