Geometry and Combinatorics - II

Let the length of major and minor axes be a and b respectively.

Area of an ellipse = $\pi a b$

Area of a circle = $\pi b^{2}$

Area outside the circle and inside the ellipse = $\pi a b - \pi b^{2}$

Probability, $P = \frac{\pi a b - \pi b^{2}}{\pi a b}$

$P = \frac{a-b}{a}$

$P = 1 - \frac{b}{a}$

Eccentricity, $e = \sqrt{1 - \frac{b^{2}}{a^{2}} }$

$\frac{b}{a} = \sqrt{1 - e^{2} }$

$P = 1 - \sqrt{1 - e^{2} }$

$P = 1 - \sqrt{1 - \frac{3^{2}}{5^{2}} }$

$P = 1 - \sqrt{1 - \frac{9}{25} }$

$P = 1 - \sqrt{\frac{16}{25} }$

$P = 1 - \frac{4}{5}$

$P = \frac{1}{5}$

$\boxed{1+5 = 6}$