I can only try-
Formulas are to a mathematician as tools are to a handyman. If a handyman doesn’t have the correct tools he’ll either go out and buy them or say he can’t do the job. Another person (who may not even know which tools are needed) will find a way to improvise.
I was always impressed at how you were able to solve problems using your own methods when I would have been at a loss if I was told that certain formulas were off limits.
With this riddle I was truly expecting you to figure out that “x” can take on values between 0 and 1/2 and “y” can take on values between 0 and 90 degrees (or 0 and ?/2 radians). The joint probability function is f(x,y) = 2 * 2 / ? = 4 / ?. A needle crosses a line if x < =(1/2) sin y.
Therefore “x” has to be between 0 and (1/2) sin “y” and “y” has to be between 0 and 2 / ?.
All of the above can be done without calculus using methods which you have figured out before on your own.
The next step requires calculus (or a numerical estimation, another technique you’ve used before). Had you gotten stuck here you could have asked me, squeak, SJSinNYC, Frummie, Feif Un… for help.
You could also find free integration software online.
Check out this ==>LINK<== to see a numerical estimation.