E and F are two independent events such that P (Ē)=0.6 and P ( E ∪ F )=0 · 6. Find P ( F ) and P (Ē ∪ F̄).

$\displaystyle \mathrm{P}(\overline{\boldsymbol{\mathrm{E}}})=0.6 \Rightarrow \boldsymbol{\mathrm{P}}(\boldsymbol{\mathrm{E}})=0.4$ $\displaystyle \mathrm{P}(\mathrm{E} \cup \mathrm{F})=\mathrm{P}(\mathrm{E})+\mathrm{P}(\mathrm{F})-\mathrm{P}(\mathrm{E} \cap \mathrm{F})$ $\displaystyle \Rightarrow 0.6=0.4+\mathrm{P}(\mathrm{F})-0.4 \mathrm{P}(\mathrm{F}) \Rightarrow \mathrm{P}(\mathrm{F})=\frac{1}{3}$ $\displaystyle \mathrm{P}(\overline{\boldsymbol{\mathrm{E}}} \cup \overline{\boldsymbol{\mathrm{F}}})=\mathbf{1}-\boldsymbol{\mathrm{P}}(\boldsymbol{\mathrm{E}} \cap \boldsymbol{\mathrm{F}})$ $$=1-0.4 \times \frac{1}{3}=\frac{13}{15} $$