What is the best shape for a soccer goal post?

[haruspex]

the distance of the point from origin will also be the distance of the line from origin as the line is perpendicular to the line joining origin and that point so you don't need to look at $d$ again since you already fixed that while picking the point

So you are saying the point you are picking in the semicircle is the point of closest approach to the origin.
That's not what you wrote in post #139. There it was the point the shot was taken from.
You cannot have it both ways. For some legal shots the point of closest approach to the origin would be behind the goal line.

 

[haruspex]

Given that S=R, the only effective difference between square and round is corner A in thread post #105, the one where the side facing the pitch meets the side facing the goalmouth.
For a shot taken from wide of the goal, it creates an obstruction, reducing the chance of a goal; for a shot taken from in front of the goal it acts as a sweeper, helping to deflect the ball into the goal.
Since the posts are relatively narrow, it is very reasonable to take d as uniformly distributed, so the major question is whether the striker is more likely to be wide of the goal or in front of it. If wide, round makes the task easier, if in front, square wins.

Edit: in practice, a striker wide of the goalmouth might aim for the far end of the goal in order to avoid the keeper. In that case square wins again.

 

[kshitij]

So you are saying the point you are picking in the semicircle is the point of closest approach to the origin.
That's not what you wrote in post #139. There it was the point the shot was taken from.
You cannot have it both ways. For some legal shots the point of closest approach to the origin would be behind the goal line.

What I wanted to do was to include all the paths of the ball that we allowed in our probability calculation, i.e., all possible straight lines whose perpendicular distance from origin is $\leq D$

But now I see that all possible paths that intersect the semi-circle of radius $D$ would not include some paths of the ball so now I say that instead of the semicircle, let us assume a complete circle.

So, I believe that now we can say that the probability that we calculated is,

"all possible paths of the ball (straight lines) that intersect a circle of radius $D$ has a probability $P(square)=\dfrac{D\pi-2S}{2D\pi}=P(round)=\dfrac{D\pi-2R}{2D\pi}$ of going in the back of the net if the other post is at a very large distance to the left of the post whose dimensions we considered"

Edit: the paths of the ball are bidirectional, i.e., for any path that intersects the circle, the direction of the shot is assumed such that it doesn't come from behind the goal line.

Edit II: the distance of paths of the ball from origin is uniformly distributed in $(-D,D)$

 

[haruspex]

all possible paths of the ball (straight lines)

But that does not specify the probability distribution. To use the result obtained earlier you need to specify that the distance from the path to the origin is uniformly distributed over the possible values.

 

[kshitij]

But that does not specify the probability distribution. To use the result obtained earlier you need to specify that the distance from the path to the origin is uniformly distributed over the possible values.

That is, add the assumption that $d$ is uniformly distributed in $(-D,D)$ then it should be fine right?

 

[kshitij]

Edit: in practice, a striker wide of the goalmouth might aim for the far end of the goal in order to avoid the keeper. In that case square wins again.

u/gegenpressing91 on reddit does some beautiful illustrations like the one below,

Here we clearly see that for wider shots, he did aim to the opposite post so your assumption is right

 

[haruspex]

That is, add the assumption that $d$ is uniformly distributed in $(-D,D)$ then it should be fine right?

Yes.

 

[kshitij]

Yes.

I would like to thank you for your patient and insightful responses throughout this thread and the other one

As mentioned earlier this was not any homework question of significance but rather something I casually made up on my own and yet you were committed to help me throughout this which I think is incredible!