The probabilities of three events A, B and C are given by P ( A )=0.6, P ( B )=0.4 and P ( C )=0.5. If P ( A ∪ B )=0.8, P ( A ∩ C )=0.3, P ( A ∩ B ∩…

The probabilities of three events A, B and C are given by $\displaystyle \mathrm{P}(\mathrm{A})=0.6, \mathrm{P}(\mathrm{B})=0.4$ and $\displaystyle \mathrm{P}(\mathrm{C})=0.5$. If $\displaystyle \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.8, \mathrm{P}(\mathrm{A} \cap \mathrm{C})=0.3$, $\displaystyle \mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0.2, \quad \mathrm{P}(\mathrm{B} \cap \mathrm{C})=\beta \quad$ and $\displaystyle \mathrm{P}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=\alpha$, where $\displaystyle 0.85 \leq \alpha \leq 0.95$, then $\displaystyle \beta$ lies in the interval :