Question

A point in a circle is selected at random. Calculate probability that point is closer to centre than circumference

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

Comments

why are you taking the area of circle??

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Because the point can be anywhere on the area occupied by the circle.

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@arjun sir,can you pls tell whts the logic of taking r/2 in favourable case?

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Draw a line from centre to the circumference- length of which will be $r$. Now, if we draw a circle with same centre and radius half, we get the location of the favorable case.

Answer 2

The probability of this is simply  π(r/2)²/πr²     =1/4  = 0.25