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Just based off of this, I think hitting the back goal is going to happen unintentionally (not saying it’s a bad thing obviously) more then people think.
We can use some basic geometry to find the approximate rate of balls that will enter the inner goal if they are evenly distributed across the outer goal area.
We shrink each target by the radius of the ball, so if the center of the ball is inside each shrunken target then the ball will enter without hitting the edge of the target. The ratio of the areas of the shrunken targets should be the approximate percentage of random shots. This doesn’t account for a number of factors, including that shots are not evenly distributed (more will be closer to the center), the inner goal is offset back into the page (so some balls will fall short), etc.
Edit: I made two mistakes with this, but they nicely cancelled each other out. We should be subtracting 2r (aka the ball diameter) from the goal sizes because they need to shrink from both sides. But also, I accidentally listed the ball’s diameter as its radius. So the final answer is still correct. I updated the drawing to replace ball radius with diameter.
