Your startup (Silicon Valley Spaghetti) is pioneering a new process for making pasta. A piece starts in machine A with probability 1/2 and in machine B with probability 1/2. The initial length of the piece is a random variable X.

If the piece starts in machine A, X has a uniform distribution on [0,1].

If the piece starts in machine B, X has a uniform distribution on [0,2].

The piece then enters the stretching machine, resulting in final length Y which is uniformly distributed on [X, X+1].

Draw two sketches:

1. A graph of the joint distribution of X and Y, conditional on machine A being selected.

2. A graph of the joint distribution of X and Y, conditional on machine B being selected.

You do not need to draw 3-dimensional plots. It is sufficient to draw the support of each joint distribution in the X-Y plane.

If the piece has final length less than 1, what is the conditional probability that it came from machine A?