Question

Find the probability that a dart hits the region X(inner circle) if the radius of the outer circle is $x$ times the radius of the inner circle.

Answer 1

Let the radius of the inner circle be $ycm$.
The radius of the outer circle will be $x\times ycm=xycm$

Let $E$ be the event of the dart hitting the region X.
The favourable outcome for $E$ will be given by the area of the innermost circle.
Since, the dart can hit anywhere on the dartboard, the total possible outcomes will be given by the area of the outermost circle.

$\therefore P\left(E\right)=\frac{\text{Area of the innermost circle}}{\text{Area of the outermost circle}}$

$\Rightarrow P\left(E\right)=\frac{\pi \times y^2}{\pi \times (x\times y{)}^2}$

$\Rightarrow P\left(E\right)=\frac{\pi \times y^2}{\pi \times x^2\times y^2}$

$\Rightarrow P\left(E\right)=\frac{1}{x^2}$

Hence, the probability that the dart hits the region X is $\frac{1}{x^2}$.