The nice answer (referred to above as a "cheat") is to note that the sought probability is the mean number of crossings made by a 1 foot needle with the lines on the floor, where we are implicitly supposing our throw-distribution to be uniform with respect to both translation and rotation. This uniformity, along with "linearity of expectation", is such that the mean number of line-crossings from throwing any shape is simply proportional to the length of the shape (imagine breaking the shape up into many tiny straight lines, all identical except for location and orientation, the number of which is proportional to the length). Note that a circle of 1 foot diameter always makes exactly 2 crossings. Thus, the constant of proportion is 2/pi per foot, and accordingly the sought probability is 2/pi.
#2 is called this: http://en.wikipedia.org/wiki/Buffon%27s_needle. This is also mentioned in Jordan Ellenberg's book How Not to Be Wrong, and has a similar passage with your explanation above. I mention the book as much because it's a great read.
The nice answer (referred to above as a "cheat") is to note that the sought probability is the mean number of crossings made by a 1 foot needle with the lines on the floor, where we are implicitly supposing our throw-distribution to be uniform with respect to both translation and rotation. This uniformity, along with "linearity of expectation", is such that the mean number of line-crossings from throwing any shape is simply proportional to the length of the shape (imagine breaking the shape up into many tiny straight lines, all identical except for location and orientation, the number of which is proportional to the length). Note that a circle of 1 foot diameter always makes exactly 2 crossings. Thus, the constant of proportion is 2/pi per foot, and accordingly the sought probability is 2/pi.