Probability of an equation being less than/greater than a certain value

probability

I'm studying probability and the form of a problem that we are working on is:

$x^2 + y^2 < n/k$, the question being what is the probability that the equation is less than (or greater than) the $n/k$, given intervals for $x$ and $y$ such as
$0 < x <1$, $-1 < y < 1$ and $n$ and $k$ are just two random values.

$n$ and $y$ are just chosen values. Distribution is not specified

A concrete example:

What is the probability that $x^2 + (1-y)^2 < 1/9$

I had started using joint density functions and integrating with respect to $x$ and $y$ using the intervals as the support for the integration limits. Doing that gets me $8/9$ but I'm not certain if that is indicative of $8/9$ < $1/9$, so the probability is just $0$ or what??? 🙂 Any help is greatly appreciated.

Thanks!

Best Answer

As question specifies no distribution, I am assuming uniform distribution.As @Did pointed out this is a badly composed problem, but since this answer solves a part of the problem. I will keep this intact.

You are looking at equation of an circle with restricted domain, This can be easily plotted as,

enter image description here

The area of the points in the domain which satisfy the condition $$x^2+(1-y)^2\lt\frac{1}{9}$$ is

$$\frac{1}{4}\pi\frac{1}{3^2}=\frac{\pi}{36}$$

The area of the domain is $2\cdot1=2$.

Assuming uniform distribution, the required probability is the ratio of these areas, that is, $$\frac{\frac{\pi}{36}}{2}=\frac{\pi}{72}$$

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