Question

a point is selected at random inside a circle what is the probability that the point is closer to the centre than the circumference is

Answer 1

Favourable outcomes: The point should be nearer to center than from circumference, it means the point could be anywhere within the radius $r/2, = \pi \times (r/2)^2$.

Total possible outcomes: The point could be anywhere within the radius $r=\pi \times r^2$.

Thus probability $= \frac{\pi \times (r/2)^2}{\pi\times r^2}= 1/4.$