# Distribution of random darts on a dart board.

If a dart thrower had the ability to throw a dart at a dart board (with radius $$r$$) and hit it every time in a random position, what would the probability density function of the distances from where the dart landed on the board to the centre?

My friend and I were arguing about what the shape of the graph would be. My friend says it would be a horizontal line, while I say otherwise. Who is correct and how would a problem like this be approached?

• The answer depends on the distribution of the darts on the board. Would you be happy to assume the darts would land uniformly over the board? Commented May 17, 2020 at 12:42
• Yes, the land uniformly over the board Commented May 17, 2020 at 12:44

We can assume the radius of the circle is $$1.$$ Let $$R$$ be a random variable denoting the distance between the center and a dart landing on the board. For any $$r\in [0,1]$$, the probability a dart lands within $$r$$ of the center is equal to the ratio of the area of the circle of radius $$r$$ to the area of the circle of radius $$1.$$ Thus we have $$\mathbb{P}(R \leq r) = \displaystyle \frac{ \pi r^2}{\pi} = r^2$$ so the probability density function is $$p_R(r) = 2r$$ for $$r\in [0,1]$$ and $$0$$ elsewhere.