# A modified Buffon's needle

A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it.

What is the probability that the needle lies between the two lines?

I can see how this would work if the length of the needle was less than the length of the lines but since it is the other way around I get a little lost. If anyone could explain this thoroughly I would appreciate it highly.

Thank you

Edit:

Well I can see that the 2 chance variables: the midpoint of the needle's least distance from a like so say that variable is x and x is between 0 and 1.125, and the other chance variable which is the rotation of the needle. Call this rotation Q and Q is between 0 and pi/2.

This is where I start to get muddy and unsure. So the needle should not cross the line if x>(1.25*Sin(Q)). So This outside integral of the double integral integrates in terms of Q from 0 to (pi/2) the inside integral I'm not sure from what to what.

The function being integrated is (16pi)/9 as it is just the probability density functions multiplied. Can anyone help me at this point?

So the angle between the needle and the lines is assumed to be uniformly distributed between $0$ and $90^\circ$. So if the needle is nearly parallel to the lines you have a high probability that it lands between them.