Question

If A and B are two events such that $P(A\cap B)=\frac{1}{3},P(A\cup B)=\frac{5}{6}\mathrm{and}P\left(\overline{\mathit{B}}\right)=\frac{1}{3}$, then P(A)=_____________.

Answer 1

For two events A and B

given P(A⋂B) = $\frac{1}{3}$

P(A⋃B) = $\frac{5}{6}$

$P\left(\overline{B}\right)=\frac{1}{3}$

$$\begin{gathered} \mathrm{i}.\mathrm{e}P\left(B\right)=1-P\left(\overline{B}\right) \\ =1-\frac{1}{3} \\ P\left(B\right)=\frac{2}{3} \end{gathered}$$
using
P(A⋃B) = P(A) + P(B) – P(A⋂B)
$\begin{array}{rcl}\frac{5}{6}& =& P\left(A\right)+\frac{2}{3}-\frac{1}{3}\\ \mathrm{i}.\mathrm{e}P\left(A\right)& =& \frac{5}{6}-\frac{2}{3}+\frac{1}{3}\\ & =& \frac{5}{6}-\frac{1}{3}\\ & =& \frac{5-2}{6}\\ & =& \frac{3}{6}\\ \mathrm{i}.\mathrm{e}P\left(A\right)& =& \frac{1}{2}\end{array}$