Question

If A and B are two independent events such that $P(A\cap B)=\frac{1}{6}$ and $P(\overline{A}\cap \overline{B})=\frac{1}{3},$ then write the values of P(A) and P(B).

Answer 1

$P(\overline{A}\cap \overline{B})=1-P(A\cup B)$

$P(A\cup B)=1-\frac{1}{3}=\frac{2}{3}P\left(A\right)+P\left(B\right)=\frac{2}{3}+\frac{1}{6}=\frac{5}{6}$ ...(i)

$\because$ A and B are two independent event

$\therefore P(A\cap B)=P\left(A\right)\times P\left(B\right)$

$=\frac{1}{6}$

${P\left(A\right)-P\left(B\right)}^2$

$=P\left(A\right)+P(B{)}^2+4P(A\left)P\right(B)$

= ${\left(\frac{5}{6}\right)}^2-\frac{4}{6}$

$=\frac{25}{36}-\frac{4}{6}$

$=\frac{25-24}{36}=\frac{1}{36}$

$\therefore P\left(A\right)-P\left(B\right)=\frac{1}{6}$ ...(ii)

From (i) and (ii)

$P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$