Is the probability of the two given distributions same?

Question

Given a PRG generating uniform random x,y coordinates. Is the probability of the given two figures same?

My contradicting arguments are:-

1. If we toss 5 coins in a row the probability of a combination is always $1/2^5$ be it HHHTT or TTHHH. Similarly the probability of a given configuration of 5 points in a 10x10 grid is $1/(10^5) * 1/(10^5)$ irrespective of where the points are.

2. Since, in the second figure, the points are only on the right half of the screen, the probability of this specific occurance is less likely.

Which of my arguments is correct?

Context: Let us say we are distributing resources using a uniform PRG. Anything on the left half of the screen is taken by A and right half is taken by B. Is figure B fair and natural or does it have a lower chance of occurring than figure A?

Answer 1

I think you're on the right track with your second argument. You could set up your null hypothesis as "The resources are distributed as a bivariate uniform distribution." You can then divide the area in half and calculate the probability of ending up with $n$ on one side and $m$ on the other. If it is very small, $<0.05$, you can reject your null hypothesis.