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જો $A$ અને $B$ એ બે સ્વત્રંત ઘટનાઓ એવી છે કે જેથી $P(A) > 0.5,\,P(B) > 0.5,\,P(A \cap \bar B) = \frac{3}{{25}},\,P(\bar A \cap B) = \frac{8}{{25}}$ થાય તો $P(A \cap B)$ ની કિમત મેળવો.
$\frac {12}{25}$
$\frac {14}{25}$
$\frac {18}{25}$
$\frac {24}{25}$
Solution
$\frac{3}{25}=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})$
$\frac{8}{25}=P(B)-P(A \cap B)=P(B)-P(A) \cdot P(B)$
$P(B)-P(A)=\frac{1}{5}$
Let $P(A)=a, P(B)=b$
$a-a b=\frac{3}{25}, b-a b=\frac{8}{25}, b-a=\frac{1}{5}$
$\therefore a-a\left(a+\frac{1}{5}\right)=\frac{3}{25}$
$a-a^{2}-\frac{a}{5}=\frac{3}{25} \Rightarrow a-5 a^{2}=\frac{3}{5}$
$\Rightarrow 25 a^{2}-20 a+3=0$
$25 a^{2}-15 a-5 a+3=0$
$\Rightarrow(5 a-1)(5 a-3)=0$
$\therefore a=3 / 5$
$\therefore b=\frac{3}{5}+\frac{1}{5}=\frac{4}{5}$
$\therefore P(A \cap B) = \frac{4}{5} \cdot \frac{3}{5} = \frac{{12}}{{25}}$
