Let we have to select point nearer to center then the circumference of a given circle.
What is the probability of finding such points?
I am confused that points at distance$\frac{r}{2}$must be favorable or not
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1The probability of what? – Hetebrij Dec 13 '16 at 13:08
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1The points at distance $r/2$ won't change anything since the probability of being exactly at that distance is zero. – msm Dec 13 '16 at 13:25
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Take a circle with centre as $O$ and radius $OA = r$ cm. Make another concentric circle OB of radius $\frac{r}{2}$ cm. Any point interior to circle with radius $r$ cm is closer to the center than to circumference. If we choose a point at random, probability that it will lie in the inner circle is :- 
$$\frac{\text{Area of Inner Circle}}{\text{Total area}} = \frac{\pi \times \frac{r}{2}\times \frac{r}{2}}{\pi\times r\times r} =\frac{1}{4}$$