ff4eca…: “Except an orange ocean”

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You pick two numbers $x$ and $y$ between $0$ and $1$ at random.

How likely is it that $x^2 + y^2 \geq 1$ ?

$\mbox {the probability of this event is } \answer {1 - \pi /4}$

Let $R = [0,1] \times [0,1]$ .

Define $S \subset R$ by $S = \{ (x,y) \in R : x^2 + y^2 \geq 1 \}$ .

The desired probability is the area of $S$ divided by the area of $R$ .

The area of $R$ is $1$ .

The area of $S$ is $\iint _S 1 \d A$ .

By geometry, the area of $S$ is $1 - \pi /4$ because $\pi /4$ is the area of a quarter of a unit circle.

So the probability is $1 - \pi /4 \approx 21\%$ .