Alice has a bag with three dice in them. In particular:

- Die 1 has the numbers 1,2,3,4,5,6 on its sides.

- Die 2 has the number 1 on four sides, and the number 2 on the remaining two sides.

- Die 3 has the number 2 on three sides, and the numbers 1,4,5 on each of the remaining three sides. Alice picks one of the three dice from the bag (each is equally likely), and then throws the die. She shows you the outcome, and the outcome only (i.e, you do not see the other faces of the die).

a) What is the probability that the outcome is 1?

b) Given the outcome is 1, what is the probability she used Die 1?

c) Given the outcome is even, what is the probability she used Die 2?

d) Consider the event A that Alice picked Die 1 and the event E that the outcome is even. Are A and E independent?