# Probability that product of distances is less than 1

Here is a probability problem I modified. Essentially, you start out with a circle of radius $$1$$, and throw a dart (we assume that you're a horrible player or that you're blindfolded so that we can neglect rotational symmetry etc...). If the dart is out of the circle, you lose. If it is in, the radius is divided by a factor numerically equal to the distance between bullseye and the point the dart landed in.

Consider the first throw. The initial radius is $$r_0 = 1$$, and the dart lands on a point with coordinates $$(x_0, y_0)$$, where the origin is centered at bullseye. Then, the next radius will be: $$r_1 = \frac{1}{\sqrt{x_0^2+y_0^2}}$$.

We then throw the second dart, which lands at a point $$(x_1,y_1)$$. Then, for this second dart to be inside the circle, we must have that: $$\sqrt{x_1^2+y_1^2} So the probability that the second dart is in is equal to the probability that $$(x_0^2+y_0^2)(x_1^2+y_1^2)<1$$ for 4 numbers $$x_0, y_0, x_1, y_1 \in [0,1]$$. My question is, what is this probability?

My thoughts are that it should be equal to the probability that the product of two numbers from 0 to 2 is less than 1. Is this correct?

EDIT: I think what we can do is: $$r_0^2r_1^2<1 \implies r_0 < \frac{1}{r_1}.$$ I can then plot $$r_0 = \frac{1}{r_1}$$ as shown below. If we divide the shaded area by the area of the square, shouldn't we get the probability we require? Desmos plot I found the shaded area to be $$2\ln2+1$$, so that the probability is $$\frac{2\ln2+1}{4}$$.

• You want $r_1 > r_0$? (If so, it's okay, but it sounds weird based on your previous comments) Commented Feb 11, 2020 at 20:52
• The radius should decrease every trial. So, $r_1<r_0=1$. EDIT: it should increase. Commented Feb 11, 2020 at 20:53
• Then you have an error in your definition of $r_1$ because it will always be greater than $1$ Commented Feb 11, 2020 at 20:55
• Ah damn sorry... slightly sleep deprived. Yeah, it should increase. Commented Feb 11, 2020 at 20:56
• But it still says "the radius is reduced"? Commented Feb 11, 2020 at 21:06