Question. Let S be the unit circle in R^2.

Let E be the collection of all Lebesgue measurable subsets of S (please

don't worry too much about this).

Let P(A) be defined as the area of A, for every subset A of S which

is measurable (i.e. a member of E - most subsets you can think of are

measurable, so don't worry about this, it's just mentioned for completeness).

Let (S, E, P) be the probability space we refer to, for this assignment

problem.

1. Give examples of three non-empty independent events.

2. Give examples of three non-empty events that are pairwise indepen-

dent but not independent.

Please give explicit mathematical descriptions of all your events. For

example, if your event is a square, you could describe it by specifying its

vertices, or the vertices on a diagonal, etc. If your event is a circle, you

should specify the coordinates of the centre and the radius.