Question

A fine dot is placed into the picture with-out looking into it. What is the probability of falling the dot in the small semicircle? What is the probability of falling the dot outside the small semicircle but inside the big semicircle?

Answer 1

Let the radius of circle be r, then the radius of

semicircle be \(\frac{r}{2},\)

Area of circle = πr²

Area of semi circle = \(\frac{1}{2}\pi(\frac{r}{2})^2=\pi\frac{r^2}{8}\)

2 x Area of semi circle = 2 x \(\pi \frac{r^2}{8}=\pi\frac{r^2}{4}\)

Area of circle without semi circle = \(\pi r^2-\frac{\pi r^2}{4}=\frac{3\pi r^2}{4}\)

∴ Probablity of the dot falling on the semi - circle = \(\frac{\frac{\pi r^2}{4}}{\frac{4}{\pi r^2}}=\frac{1}{4}\)

Probablity of the dot falling inside of any semicircle is = \(\frac{\frac{\pi r^2}{8}}{\pi r^2}=\frac{1}{8}\)

Probablity of the dot falling outside of semicircle = \(\frac{\frac{3\pi r^2}{4}}{\pi r^2}=\frac{3}{4}\)